A right R-module M is said to be a test module for flatness (shortly: an f-test module) provided for each left R-module N, Tor(M,N) = 0 implies N is flat. f-test modules are a flat version of the Whitehead test modules for injectivity deflned by Trlifaj. In this paper the properties of f-test modules are investigated and are used to characterize various families of rings. The structure of a ring over which every (flnitely generated) right R-module is flat or f-test is investigated. Abelian groups that are Whitehead test modules for injectivity or f-test are characterized.