  There are four non-isomorphic configurations of triples that can form a
triangle in a $3$-uniform hypergraph. Forbidding different combinations of
these four configurations, fifteen extremal problems can be defined, several of
which already appeared in the literature in some different context. Here we
systematically study all of these problems solving the new cases exactly or
asymptotically. In many cases we also characterize the extremal constructions.
