Abstract: Symmetry plays a central role in the sciences, machine learning, and statistics. When data are known to obey a symmetry, various methods that exploit symmetry have been developed. However, statistical tests for the presence of group invariance largely focus on a handful of specialized situations, and tests for equivariance are largely non-existent. This work formulates non-parametric hypothesis tests, based on a single independent and identically distributed sample, for distributional symmetry under a specified group. We provide a general formulation of tests for symmetry within two broad settings. The first setting tests for the invariance of a marginal or joint distribution under the action of a compact group. We propose a conditional Monte Carlo test that achieves exact $p$-values with finitely many observations and Monte Carlo samples. The second setting tests for the invariance or equivariance of a conditional distribution under the action of a locally compact group. We show that the test for conditional symmetry can be formulated as a test of conditional independence. We implement our tests using kernel methods and apply them to testing for symmetry in problems from high-energy particle physics.