##Eberhard Theorem
Consider a simple 3-polytope P, (so every vertex has 3 neighbors). If $p_k$ is the number of faces of P which are k-gonal, Euler's theorem implies that
$(*) \sum_{k \ge 3} (6-k)p_k=12$.
Note that 6-gonal faces do not contribute to the LHS. (One way to think about it is that polygonal faces with 7 and more sides contributes "negative curvature", small faces contributes positive "curvature" and hexagons are "flat".)
Eberhard's theorem asserts that if you have a sequence of numbers $p_k, k \ne 6$ such that $\sum_{k \ge 3} (6-k)p_k=12$ then you can find a simple 3-polytope with $p_k$ k-gonal faces. (But you have no control on $p_6$).
##Extensions of Eberhard theorem
There are various results extending Eberhard's theorem in various directions. Chapter 13 in Grunbaum's book "Convex Polytopes" and especially the supplemantary material at the end of the chapter in the new 2nd edition is a good source. Another general source is my chapter from the "Handbook of Discrete and Computational geometry" on garphs and skeleta of polytops.
A relatively recent paper on the subject is by Stanislav Jendrol "On the face vectors of trivalent convex polyhedra". Another paper by Jendrol which deals with general 3-polytopes from the same year is "On face vectors and vertex vectors of convex polyhedra" Discrete Math 118 (1993)119-144. There are analogs of Eberhard theorem for 4-regular planar graphs, for toroidal graphs and in other directions.
A far as I know, there is no good answer known for the question posed by Shephard of characterizing all sequences $(p_3,p_4,\dots)$, and no such characterization is known even for the simple case.
##High dimensions
In higher dimensions and even in four dimensions there are various different ways to extend these problems, these problems become very difficult and very little is known.
###2-dimensional faces
You can ask again about the numbers of k-gonal 2-dimensional faces. While the formula above implies that in dimension 3 and more $p_3+p_4+p_5>0$ it is known that in dimensions 5 and more $p_3+p_4>0$.
Types of facets
Perhaps an even more natural extension is to consider the type of facets a given d-simensional polytope have. You can ask for a simple 4 polytope (a 4-polytope whose graph is 4-regular) what are the number of facets $p_Q$ isomorphic to a given 3-polytope Q. This gives you a vector indexed by combinatorial types of simple 3-polytopes, but I am not aware of any Eberhard type theorem and I do not know even which 3-polytopes should be considered as the analogs of the hexagons in the above formula. Dually stated and extented to triangulations of 3-spheres the question is to associate to a triangulated 3-simensional sphere (or just simplicial 3-polytope) the list of links of vertices (with multiplicities) it has. A related MO question is this one.
Type of facets according to their number of their facets
Rather than classifying the facets according to their combinatorial type we can classify them according to their own number of facets. Here, for dimension greater than 4 there is no analog for (*). It is possible that under wide circumstences the numbers of facets with k facets can be prescribed, but I am not aware of results in this directio.
Type of facets according to their f-vectors
Precscribing the entire vector of face numbers of the individual facets, may well be the most interesting extension from 3 to higher dimensions.
There are some reasons to jump from dimension 3 directly to dimension 5. The nature of the problem is different in even and odd dimensions. I will try to elaborate on that in the next edit.