  Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs
$\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the
minimum number of colors needed to edge-color $G$ so that every copy of every
hypergraph $H\in \mathcal{H}$ in $G$ receives at least $q$ different colors. In
this note we obtain bounds, some asymptotically sharp, on several generalized
Ramsey numbers, when $G=K_n$ or $G=K_{n,n}$ and $\mathcal{H}$ is a set of
cycles or paths, and when $G=K_n^k$ and $\mathcal{H}$ contains a clique on
$k+2$ vertices or a tight cycle.
