I will survey what is known concerning the basic behavior of random walks on finitely generated groups in the simple case where the probability measure driving the walk is finitely supported, symmetric and non-degenerate. Here, basic behavior refers to the behavior of the probability of return to the starting point. For those finitely generated groups that can be realized as a closed subgroup of a Lie group, the possible behaviors are classified into three types (classical behaviors). However, many other behaviors are possible even in the class of solvable groups.