A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results shows that for every $ε>0$ there exists $n_0$ such that if $n>n_0$ and $\frac25+ε<\frac kn<\frac 12-ε$ then the maximum size is $4\binom{n-4}{k-3}+\binom{n-4}{k-4}$.