Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman
investigated the problem of maximizing the number of T copies in an H-free
graph, for a pair of graphs T and H. Whereas Alon and Shikhelman were
primarily interested in determining the order of magnitude for large classes of
graphs H, we focus on the case when T and H are paths, where we find
asymptotic and in some cases exact results. We also consider other structures
like stars and the set of cycles of length at least k, where we derive
asymptotically sharp estimates. Our results generalize well-known extremal
theorems of Erd\H{o}s and Gallai