  We introduce a notion of $k$-convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{$k$-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{$O(n
\log n)$} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{$2$-convex} objects considered.
