Originally motivated by a problem of Erdős on the maximum number of maximum-area triangles determined by an $n$-point set, we will discuss some new and old results on the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact by sharing and/or interleaving vertices.