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 complex, decides whether it embeds in d-dimensional space? (2) What is the maximum number of k-dimensional simplices of a simplicial complex that embeds into d-dimensional space? Time permitting, we will also discuss some maps with more general restrictions on the set of singularities, e.g., without r-fold intersection points. The talk will be of a survey-like nature and touch upon joint work with a number of colleagues: M. Cadek, X. Goaoc, M. Krcal, I. Mabillard, J. Matousek, P. Patak, Z. Safernova, F. Sergeraert, E. Sedgwick, M. Tancer, and L. Vokrinek