It is a standard result of elementary homological algebra that to every Rmodule $A$ there exists a projective resolution. It is often said that the category of Rmodules has "enough projectives." In which other categories is this also true? In particular is it true for abelian categories?
Among the standard examples of abelian categories without enough projectives, there are
No abelian category where the functors of infinite product are not exact can have enough projectives. In Grothendieck categories (i.e. abelian categories with exact functors of small filtered colimits and a set of generators) there are always enough injectives, but may be not enough projectives. Among the standard examples of abelian categories with enough projectives, there are



Way too optimistic: in many abelian categories there are not enough projectives (and in the dual category there are not enough injectives). The most standard example is sheaves of abelian groups on a topological space X. For most X, this category does not have enough projectives. See for example this question where this was discussed: When are there enough projective sheaves on a space X? On the positive side: if A has enough projectives and I is a small category then the category $A^I$ (i.e. the category of functors $F:I\to A$) has enough projectives (assuming arbitrary sums exist in A). In particular, the category of complexes in A has enough projectives (taking $I=\mathbb Z$ with arrows $d_n:n\to n+1$ satisfying $d_{n+1}\circ d_n=0$). See this question: How to construct pair of adjoint functors from category A to category A_D(category of diagrams) All of these are abelian categories. 


A simple example of an abelian category having not enough projectives is the category of finite abelian groups. In fact, it contains neither nontrivial projective objects nor nontrivial injective objects. 

