What are the possible face numbers of d-dimensional: polytopes? simplicial polytopes (namely those whose proper faces are all simplices)? flag polytopes (namely simplicial polytopes all whose minimal nonfaces have size 2)? centrally symmetric polytopes (namely those invariant under the map x-->-x)?...

We'll survey results and open problems concerning face numbers (f-vectors) of polytopes and simplicial complexes, including subfamilies of particular interest.

As the area is vast and fast growing, this talk will cover only a portion of it, representing some of the main methods used. These methods range from algebraic to combinatorial, topological and geometric, and will form Part II given by Martina Juhnke-Kubitzke.