Let \(\mathfrak V\) be a variety of groups defined by a set of laws \(\mathcal V\). An element \(w\) of a group \(G\) is called \(\mathfrak V\)-generic in \(G\) if \(w\) is in \({\mathcal V} (G)\), the verbal subgroup of \(G\) corresponding to \(\mathcal V\), and if whenever \(\alpha : H \to G\) is a group homomorphism with \(w = (u)\alpha\) for some \(u \in {\mathcal V} (H)\), then \(\alpha\) is surjective. We show that if the relatively free groups of \(\mathfrak V\) have solvable word problem, one can decide which elements of a free group are \(\mathfrak V\)- generic. We give examples of elements of free groups which are generic for the varieties \(\mathfrak A_n\) of abelian groups with exponent dividing \(n\) and the variety \({\mathfrak N}_2\) of nilpotent groups of class at most two. We also show that there is an algorithm to identify elements of free groups with the property that every endomorphism that fixes the element is an automorphism; such elements are called test elements.