  Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space}
of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially
equivalent to~$P$, modulo affine transformations. We report on work by the
first author, which shows that realization spaces of \mbox{4-dimensional}
polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic
set~$V$ defined over~$\Z$, there is a $4$-polytope $P(V)$ whose realization
space is ``stably equivalent'' to~$V$. This implies that the realization space
of a $4$-polytope can have the homotopy type of an arbitrary finite simplicial
complex, and that all algebraic numbers are needed to realize all $4$-
polytopes. The proof is constructive. These results sharply contrast the
$3$-dimensional case, where realization spaces are contractible and all
polytopes are realizable with integral coordinates (Steinitz's Theorem). No
similar universality result was previously known in any fixed dimension.
