  We consider a variation of the classical Erd\H{o}s-Szekeres problems on the
existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set
of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show
bounds and structural results on maximizing and minimizing their numbers. Most
noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower
bound for the number of $k$-holes, and show that this number is maximized by
sets in convex position.
