We initiate a thorough study of distributed property testing—producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called dense graph testing model we emulate sequential tests for nearly all graph properties having 1-sided tests, while in the general model we obtain faster tests for triangle-freeness and cycle-freeness, and in the sparse model we obtain a faster test for bipartiteness. In addition, we show a logarithmic lower bound for testing bipartiteness and cycle-freeness, which holds even in the stronger LOCAL model. In most cases, aided by parallelism, the distributed algorithms have a much shorter running time than their counterparts from the sequential querying model of traditional property testing. More importantly, the distributed algorithms we develop for testing graph properties are in many cases much faster than what is known for exactly deciding whether the property holds. The simplest property testing algorithms allow a relatively smooth transition to the distributed model. For the more complex tasks we develop new machinery that may be of independent interest. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.