  Understanding how the cycles of a graph or digraph behave in general has
always been an important point of graph theory. In this paper, we study the
question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles)
of distinct lengths in a given graph (resp. digraph). In the context of
undirected graphs, we prove that, for every $k \geq 1$, every graph with
minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of
different lengths, where the degree bound is best possible. We also consider
stronger situations, and exhibit degree bounds (some of which are best
possible) when e.g. the graph is triangle-free, or the $k$ cycles are requested
to have different lengths congruent to some values modulo some $r$. In the
context of directed graphs, we consider a conjecture of Lichiardopol concerning
the least minimum out-degree required for a digraph to have $k$ vertex-disjoint
directed cycles of different lengths. We verify this conjecture for
tournaments, and, by using the probabilistic method, for regular digraphs and
digraphs of small order.
