  The anti-Ramsey problem was introduced by Erd\H{o}s, Simonovits and S\'{o}s
in 1970s. The anti-Ramsey number of a hypergraph $\mathcal{H}$, $ar(n,s,
\mathcal{H})$, is the smallest integer $c$ such that in any coloring of the
edges of the $s$-uniform complete hypergraph on $n$ vertices with exactly $c$
colors, there is a copy of $\mathcal{H}$ whose edges have distinct colors. In
this paper, we determine the anti-Ramsey numbers of linear paths and loose
paths in hypergraphs for sufficiently large $n$, and give bounds for the
anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are
obtained for linear/loose cycles, and bounds are obtained for Berge cycles. Our
main tools are path extension technique and stability results on hypergraph
Tur\'{a}n problems of paths and cycles.
