  We survey some of the known results on the relation between the homology of
the {\em full} Hecke algebra of a reductive $p$-adic group $G$, and the
representation theory of $G$. Let us denote by $\CIc(G)$ the full Hecke algebra
of $G$ and by $\Hp_*(\CIc(G))$ its periodic cyclic homology groups. Let $\hat
G$ denote the admissible dual of $G$. One of the main points of this paper is
that the groups $\Hp_*(\CIc(G))$ are, on the one hand, directly related to the
topology of $\hat G$ and, on the other hand, the groups $\Hp_*(\CIc(G))$ are
explicitly computable in terms of $G$ (essentially, in terms of the conjugacy
classes of $G$ and the cohomology of their stabilizers). The relation between
$\Hp_*(\CIc(G))$ and the topology of $\hat G$ is established as part of a more
general principle relating $\Hp_*(A)$ to the topology of $\Prim(A)$, the
primitive ideal spectrum of $A$, for any finite typee algebra $A$. We provide
several new examples illustrating in detail this principle. We also prove in
this paper a few new results, mostly in order to better explain and tie
together the results that are presented here. For example, we compute the
Hochschild homology of $\maO(X) \rtimes \Gamma$, the crossed product of the
ring of regular functions on a smooth, complex algebraic variety $X$ by a
finite group $\Gamma$. We also outline a very tentative program to use these
results to construct and classify the cuspidal representations of $G$. At the
end of the paper, we also recall the definitions of Hochschild and cyclic
homology.
