We discuss a number of open questions and results concerning
(from a combinatorialist's point of view) higher-dimensional
analogues of graph planarity and crossing numbers, i.e., embeddings
of finite simplicial complexes (compact polyhedra) into Euclidean space.
While embeddings are a classical topic in geometric topology, here
we focus rather on algorithmic and combinatorial aspects. Two typical
questions are the following:
(1) Is there an algorithm that, given as input a finite k-dimensional simplical