A general definition of a weakly mixing action of a group is given and many alternative characterizing properties of weakly mixing unitary representations are given. A category result shows that on the unitary level, weakly mixing actions are residual for amenable groups (or generally groups in which all continuous positive definite functions can be uniformly approximated on compacta by continuous positive definite functions vanishing at infinity). Examples of special groups and properties of their actions due to the representations of the group are discussed. The abstract theory is summarized for actions induced by groups of measure-preserving transformations of a probability space.