In this paper, we study $r$-uniform hypergraphs ${\cal H}$ without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for $r = 3$, we show that if ${\cal H}$ has $n$ vertices and a maximum number of edges, then $$|{\cal H}|={\textstyle 1\over6}n^{3/2} + o(n^{3/2}).$$ This also asymptotically determines the generalized Turán number $T_{3}(n,8,4)$. Some results are based on our bounds for the maximum size of Sidon-type sets in $\Bbb{Z}_{n}$.