  We introduce combinatorial types of arrangements of convex bodies, extending
order types of point sets to arrangements of convex bodies, and study their
realization spaces. Our main results witness a trade-off between the
combinatorial complexity of the bodies and the topological complexity of their
realization space. First, we show that every combinatorial type is realizable
and its realization space is contractible under mild assumptions. Second, we
prove a universality theorem that says the restriction of the realization space
to arrangements polygons with a bounded number of vertices can have the
homotopy type of any primary semialgebraic set.
