  Fix a $k$-chromatic graph $F$. In this paper we consider the question to
determine for which graphs $H$ does the Tur\'an graph $T_{k-1}(n)$ have the
maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$
large enough). We say that such a graph $H$ is $F$-Tur\'an-good. In addition to
some general results, we give (among others) the following concrete results:
  (i) For every complete multipartite graph $H$, there is $k$ large enough such
that $H$ is $K_k$-Tur\'an-good.
  (ii) The path $P_3$ is $F$-Tur\'an-good for $F$ with $\chi(F) \geq 4$.
  (iii) The path $P_4$ and cycle $C_4$ are $C_5$-Tur\'an-good.
  (iv) The cycle $C_4$ is $F_2$-Tur\'an-good where $F_2$ is the graph of two
triangles sharing exactly one vertex.
