(Co)homology of the Eilenberg-MacLane spaces K(G,n) Let $(G, n)$ be a pair such that $n$ is a natural number, $G$ is a finite group which is abelian if $n \geq 1$. It is well-known that $\pi_n(K(G,n)) = G$ and $\pi_i (K(G,n)) = 0$ if $i \neq n$.
Also it is known that these spaces $K(G,n)$ play a very important role for cohomology. For any abelian group $G$, and any CW-complex $X$, the set $[X, K(G,n)]$ of homotopy classes of maps from $X$ to $K(G,n)$ is in natural bijection with the $n^{\mathrm{th}}$ singular cohomology group $H^n(X; G)$ with coefficients in $G$.
But what is known about the cohomology of the $K(G,n)$ themselves? It is interesting in the light of the above. Here I mean the singular cohomology with integral coefficients. 
 A: In Q-subalgebras, Milnor basis, and cohomology
of Eilenberg – Mac Lane spaces Tamanoi gives explicit polynomial generators of $H^*(K(\mathbb Z/p^k,n);\mathbb Z/p)$ and $H^*(K(\mathbb Z,n);\mathbb Z/p)$ for all primes $p$, using Milnor basis of the dual Steenrod algebra.
A: Computing the integral cohomology of $K(\pi,n)$'s is feasible but a bit tricky. In fact the only reference I know is exposé 11 of H. Cartan's seminar, year 7. I'd be interested if there are other sources that cover that.
A: You might find some useful information in the Lausanne thesis of Alain Clément:
http://doc.rero.ch/record/482/files/Clement_these.pdf
In particular, he gives an account of Cartan's results in Chapter 2, then describes a C++ program for computing integral (co)homology of certain ($2$-local) Eilenberg-Mac Lane spaces in Chapter 3. An appendix lists the integral (co)homology groups of $K(\mathbb{Z}_2,2)$, $K(\mathbb{Z}_2,3)$, $K(\mathbb{Z}_4,2)$ and $K(\mathbb{Z}_4,3)$ up to degree $200$.
A: Expanding slightly on Ryan's comment: it's an easy fact (often attributed to Serre) that 
the set of cohomology operations $H^k(-;G)\to H^r(-;H)$ (i.e. natural transformations) is in 1-1 correspondence with $[K(G,k),K(H,r)]=H^r(K(G,k);H)$ (for any abelian groups $G,H$). 
There are tons of these, some easy, some not so easy to understand, corresponding to how easy the calculation of $H^r(K(G,k);H)$ is. A nice subset are the stable operations (compatible with a certain suspension $[K(G,k),K(H,r)]\to [K(G,k+1),K(H,r+1)]$  which come in families, the most common family being the Steenrod algebra, corresponding to $G=H=Z/p$. There are non-stable operations also, eg the Pontryagin square $H^k(-;Z/2)\to H^{2k}(-;Z/4)$.
A: If you don't mind a little notational pain, you can look at the original papers where Eilenberg and Mac Lane worked many cases out: On the homology of groups $H(\pi,n)$ (I, II, III).  Annals of Mathematics ~1953.
