diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index 878508f3058cc8e163ac276631b7336800545fad..6b4d76de1f850cb5c1c8866adb358a7ac7661593 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -23,6 +23,29 @@ jobs:
                 curl -L -O -k https://gitlab.tiker.net/inducer/ci-support/raw/main/prepare-and-run-flake8.sh
                 . ./prepare-and-run-flake8.sh "$(basename $GITHUB_REPOSITORY)" examples test
 
+    pylint:
+        name: Pylint
+        runs-on: ubuntu-latest
+        steps:
+        -   uses: actions/checkout@v2
+        -   name: "Main Script"
+            run: |
+                sudo apt update
+                sudo apt install openmpi-bin libopenmpi-dev
+
+                CONDA_ENVIRONMENT=.test-conda-env-py3.yml
+                echo "- mpi4py" >> $CONDA_ENVIRONMENT
+                echo "- scipy" >> $CONDA_ENVIRONMENT
+                echo "-------------------------------------------"
+                cat $CONDA_ENVIRONMENT
+                echo "-------------------------------------------"
+                USE_CONDA_BUILD=1
+
+                curl -L -O -k https://gitlab.tiker.net/inducer/ci-support/raw/master/prepare-and-run-pylint.sh
+
+                . ./prepare-and-run-pylint.sh "$(basename $GITHUB_REPOSITORY)" test/*.py \
+                    $(find examples -name '*.py')
+
     pytest3:
         name: Pytest on Py3
         runs-on: ubuntu-latest
diff --git a/.gitlab-ci.yml b/.gitlab-ci.yml
index b0f0aa6ce26727c33f0820c68cbc204c81976fd9..a0f265290e65a39f10f6a9c36193278817c3d038 100644
--- a/.gitlab-ci.yml
+++ b/.gitlab-ci.yml
@@ -106,3 +106,14 @@ Flake8:
   - python3
   except:
   - tags
+
+Pylint:
+  script: |
+    EXTRA_INSTALL="pybind11 make numpy scipy matplotlib mpi4py"
+    curl -L -O -k https://gitlab.tiker.net/inducer/ci-support/raw/master/prepare-and-run-pylint.sh
+    . ./prepare-and-run-pylint.sh "$CI_PROJECT_NAME" test/*.py \
+        $(find examples -name '*.py')
+  tags:
+  - python3
+  except:
+  - tags
diff --git a/.pylintrc-local.yml b/.pylintrc-local.yml
new file mode 100644
index 0000000000000000000000000000000000000000..be1948583b86a81c56b102f4c159e1282e5d31dd
--- /dev/null
+++ b/.pylintrc-local.yml
@@ -0,0 +1,11 @@
+- arg: ignore
+  val:
+    - mappers
+    - gas_dynamics
+    - burgers.py
+    - diffusion.py
+    - dt_finding.py
+    - nd_calculus.py
+    - pml.py
+    - poisson.py
+    - second_order.py
diff --git a/examples/dagrt-fusion.py b/examples/dagrt-fusion.py
index 779a783cc991f5a5be1979142c1d45a2bd27789b..436d5c9d026b417ced96ed77e1a4afe2a44ee330 100755
--- a/examples/dagrt-fusion.py
+++ b/examples/dagrt-fusion.py
@@ -1294,8 +1294,7 @@ def main():
     else:
         if not SKIP_TESTS:
             # Run tests.
-            from py.test import main
-            result = main([__file__])
+            result = pytest.main([__file__])
             assert result == 0
 
         # Run examples.
diff --git a/grudge/discretization.py b/grudge/discretization.py
index 125f2ca7c4a0dcdec6f163c4df2abecff79ea161..a15f43736f5e784ccbc7e7a2a00cb90cf335b544 100644
--- a/grudge/discretization.py
+++ b/grudge/discretization.py
@@ -110,7 +110,7 @@ class DiscretizationCollection:
             return True
         else:
             return self.mpi_communicator.Get_rank() \
-                    == self._get_management_rank_index()
+                    == self.get_management_rank_index()
 
     def _set_up_distributed_communication(self, mpi_communicator, array_context):
         from_dd = sym.DOFDesc("vol", sym.QTAG_NONE)
@@ -383,9 +383,7 @@ class DiscretizationCollection:
 
     def is_volume_where(self, where):
         from grudge import sym
-        return (
-                where is None
-                or where == sym.VTAG_ALL)
+        return where is None or sym.as_dofdesc(where).is_volume()
 
     @property
     def order(self):
diff --git a/grudge/eager.py b/grudge/eager.py
index 32eccaaa99f4893873db6465c2ba249db2d353b8..22dd4298af53c277168f4d5c3203bd6e879e3b4b 100644
--- a/grudge/eager.py
+++ b/grudge/eager.py
@@ -60,7 +60,7 @@ class EagerDGDiscretization(DiscretizationCollection):
         return op.local_d_dx(self, xyz_axis, vec)
 
     def div(self, vecs):
-        return op.local_d_dx(self, vecs)
+        return op.local_div(self, vecs)
 
     def weak_grad(self, *args):
         return op.weak_local_grad(self, *args)
diff --git a/grudge/models/__init__.py b/grudge/models/__init__.py
index 6b2d4e59e8c503d123a1790112afc9ce7c1eeef1..8486bd2e059f0026081cc88a90bb258a5a08f9d5 100644
--- a/grudge/models/__init__.py
+++ b/grudge/models/__init__.py
@@ -48,6 +48,9 @@ class TimeDependentOperator(Operator):
 class HyperbolicOperator(Operator):
     """A base class for hyperbolic Discontinuous Galerkin operators."""
 
+    def max_eigenvalue(self, t, fields, discr):
+        raise NotImplementedError
+
     def estimate_rk4_timestep(self, discr, t=None, fields=None):
         """Estimate the largest stable timestep for an RK4 method.
         """
diff --git a/grudge/models/em.py b/grudge/models/em.py
index d8fbdb4ee0d967b8531530c9c6ff8c1752151f03..244d324e0749ef48d19a67af9ab0be5c3156aa0c 100644
--- a/grudge/models/em.py
+++ b/grudge/models/em.py
@@ -323,7 +323,9 @@ class MaxwellOperator(HyperbolicOperator):
             return 1/sqrt(self.epsilon*self.mu)  # a number
         else:
             import grudge.symbolic as sym
-            return sym.NodalMax()(1/sym.FunctionSymbol("sqrt")(self.epsilon*self.mu))
+            return sym.NodalMax("vol")(
+                    1 / sym.sqrt(self.epsilon * self.mu)
+                    )
 
     def max_eigenvalue(self, t, fields=None, discr=None, context={}):
         if self.fixed_material:
diff --git a/grudge/models/gas_dynamics/__init__.py b/grudge/models/gas_dynamics/__init__.py
index 8c86c8e0aed0c4cae043cec274f171122668f4cc..7562bae3148bdf6e4d3833a737b4939c52a595ff 100644
--- a/grudge/models/gas_dynamics/__init__.py
+++ b/grudge/models/gas_dynamics/__init__.py
@@ -661,7 +661,7 @@ class GasDynamicsOperator(TimeDependentOperator):
                 result = RestrictToBoundary(tag)(self.sensor())
                 return cse(to_bdry_quad(result), "bdry_sensor")
 
-        from grudge.symbolic import SubstitutionMapper
+        from grudge.symbolic.mappers import SubstitutionMapper
         return SubstitutionMapper(subst_func)(expr)
 
     # }}}
diff --git a/grudge/models/wave.py b/grudge/models/wave.py
index a06295bb4743c671c9ce33c5bd00258733f96923..5fc65a70e2235ea03d33ef95ad7e91e302f313c4 100644
--- a/grudge/models/wave.py
+++ b/grudge/models/wave.py
@@ -110,7 +110,7 @@ class WeakWaveOperator(HyperbolicOperator):
             warn("Inhomogeneous Dirichlet conditions on the wave equation "
                     "are still having issues.")
 
-            dir_g = sym.Field("dir_bc_u")
+            dir_g = sym.var("dir_bc_u")
             dir_bc = flat_obj_array(2*dir_g - dir_u, dir_v)
         else:
             dir_bc = flat_obj_array(-dir_u, dir_v)
@@ -254,7 +254,7 @@ class VariableCoefficientWeakWaveOperator(HyperbolicOperator):
             warn("Inhomogeneous Dirichlet conditions on the wave equation "
                     "are still having issues.")
 
-            dir_g = sym.Field("dir_bc_u")
+            dir_g = sym.var("dir_bc_u")
             dir_bc = flat_obj_array(dir_c, 2*dir_g - dir_u, dir_v)
         else:
             dir_bc = flat_obj_array(dir_c, -dir_u, dir_v)
@@ -310,7 +310,7 @@ class VariableCoefficientWeakWaveOperator(HyperbolicOperator):
             self.radiation_tag])
 
     def max_eigenvalue(self, t, fields=None, discr=None):
-        return sym.NodalMax()(sym.FunctionSymbol("fabs")(self.c))
+        return sym.NodalMax("vol")(sym.fabs(self.c))
 
 # }}}
 
diff --git a/grudge/op.py b/grudge/op.py
index a9a514e5ffdb415ab0359d07a76815d2dcbfe22f..b8204cf16e9254f0df68c1d8781b94914b165376 100644
--- a/grudge/op.py
+++ b/grudge/op.py
@@ -364,7 +364,7 @@ def norm(dcoll, vec, p, dd=None):
     if isinstance(vec, np.ndarray):
         if p == 2:
             return sum(
-                    norm(dcoll, vec[idx], dd=dd)**2
+                    norm(dcoll, vec[idx], p, dd=dd)**2
                     for idx in np.ndindex(vec.shape))**0.5
         elif p == np.inf:
             return max(
diff --git a/grudge/symbolic/mappers/__init__.py b/grudge/symbolic/mappers/__init__.py
index fcbb89ea3e0645b3e6b30fd6a5b83089862fa89b..799a0bc5e1a311a54a17309a746121946f86309c 100644
--- a/grudge/symbolic/mappers/__init__.py
+++ b/grudge/symbolic/mappers/__init__.py
@@ -502,7 +502,7 @@ class OperatorSpecializer(CSECachingMapperMixin, IdentityMapper):
             IdentityMapper.map_common_subexpression
 
     def map_operator_binding(self, expr):
-        from grudge.symbolic.primitives import BoundaryPair
+        from grudge.symbolic.primitives import TracePair
 
         from grudge.symbolic.mappers.type_inference import (
                 type_info, QuadratureRepresentation)
@@ -520,7 +520,7 @@ class OperatorSpecializer(CSECachingMapperMixin, IdentityMapper):
                 field_repr_tag = field_type.repr_tag
             except AttributeError:
                 # boundary pairs are not assigned types
-                assert isinstance(expr.field, BoundaryPair)
+                assert isinstance(expr.field, TracePair)
                 has_quad_operand = False
             else:
                 has_quad_operand = isinstance(field_repr_tag,
diff --git a/grudge/symbolic/operators.py b/grudge/symbolic/operators.py
index 5a48a12a8551757713af6413cfc92143a0a1857d..b6c385b1612f87e42aa4ceb0a5beb6704cb14193 100644
--- a/grudge/symbolic/operators.py
+++ b/grudge/symbolic/operators.py
@@ -424,12 +424,11 @@ class FilterOperator(ElementwiseLinearOperator):
             for mid in ldis.generate_mode_identifiers()]
 
         # build filter matrix
+        import numpy.linalg as la
         vdm = ldis.vandermonde()
-        from grudge.tools import leftsolve
         mat = np.asarray(
-            leftsolve(vdm,
-                np.dot(vdm, np.diag(filter_coeffs))),
-            order="C")
+                la.solve(vdm.T, np.diag(filter_coeffs) @ vdm.T).T,
+                order="C")
 
         return mat
 
@@ -448,7 +447,7 @@ class AveragingOperator(ElementwiseLinearOperator):
         if dd_in != dd_out:
             raise ValueError("dd_in and dd_out must be identical")
 
-        super(FilterOperator, self).__init__(dd_in, dd_out)
+        super().__init__(dd_in, dd_out)
 
     def matrix(self, eg):
         # average matrix, so that AVE*fields = cellaverage(fields)
diff --git a/grudge/symbolic/tools.py b/grudge/symbolic/tools.py
index 12d8c47fe66a473e2d380c83e3ea09c95b0dd1aa..7907060ce3bc5a64642805d22976f4c5e1096348 100644
--- a/grudge/symbolic/tools.py
+++ b/grudge/symbolic/tools.py
@@ -58,7 +58,7 @@ def split_sym_operator_for_multirate(state_vector, sym_operator,
 
         killers.append(IndexGroupKillerSubstMap(kill_set))
 
-    from grudge.symbolic import \
+    from grudge.symbolic.mappers import \
             SubstitutionMapper, \
             CommutativeConstantFoldingMapper
 
diff --git a/test/mesh_data.py b/test/mesh_data.py
index 14cee46d54aa1fbb823479d51ede6c5c1e419fe2..fbe2712fc70ecc811aae9721a62831cb09ff30df 100644
--- a/test/mesh_data.py
+++ b/test/mesh_data.py
@@ -31,7 +31,7 @@ class Curve2DMeshBuilder(MeshBuilder):
     def get_mesh(self, resolution, mesh_order):
         from meshmode.mesh.generation import make_curve_mesh
         return make_curve_mesh(
-                self.curve_fn,
+                self.curve_fn,      # pylint: disable=no-member
                 np.linspace(0.0, 1.0, resolution + 1),
                 mesh_order)
 
diff --git a/test/test_grudge.py b/test/test_grudge.py
index edbdc10362b300a4bfc14a3d894a02a276d3f996..8401caf5b87d53adb68c062250ab8991aaa641cf 100644
--- a/test/test_grudge.py
+++ b/test/test_grudge.py
@@ -166,7 +166,7 @@ def _ellipse_surface_area(radius, aspect_ratio):
         # NOTE: hardcoded value so we don't need scipy for the test
         ellip_e = 1.2110560275684594
     else:
-        from scipy.special import ellipe
+        from scipy.special import ellipe        # pylint: disable=no-name-in-module
         ellip_e = ellipe(eccentricity)
 
     return 4.0 * radius * ellip_e