We initiate the study of the space complexity of property testing. We focus on five properties of Boolean functions that have been intensively studied in the property testing literature: monotonicity, symmetry, linearity, being a junta, and being a linear threshold function.

On the upper bounds side, we show --- perhaps surprisingly --- that for most of these properties there exist new testing algorithms that use dramatically less space than the well-known testers for these properties, which have been designed to use as few queries as possible. For each of these five properties we also prove an unconditional space lower bound for any tester, regardless of its runtime or query complexity, that either matches or almost matches the space used by our new testers. Our results thus give an essentially complete characterization of the space complexity of testing these fundamental properties.