This thesis has four parts. In the first part, we introduce and study representation homology of topological spaces, which is a higher homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology in terms of classical (abelian) homological algebra. Our construction is parallel to Pirashvili's construction of higher Hochschild homology; in fact, we establish a direct relation between the two theories by proving that the representation homology of the (reduced) suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other known homology theories associated with spaces (such as the Pontryagin algebra $ H_*(\Omega X) $, the $S^1$-equivariant homology of the free loop space and the stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly in a number of interesting cases, including the spheres $S^n$, the complex projective spaces $CP^r $, closed surfaces of arbitrary genus and some 3-dimensional manifolds, such as link complements in $R^3$ and lens space $L(p,q)$. One of our main results, which we call Comparison Theorem, expresses the representation homology of a simply-connected topological space of finite rational type in terms of its Quillen and Sullivan models. The second part is a compendium of the first part. We prove some technical results required to establish some basic properties of representation homology. A result that might be of independent interest is Theorem 10.3.2, which says that if $k$ is a field of characteristic zero, then for any $k$-linear operad $P$, the model category of simplicial $P$ algebras is Quillen equivalent to the model category of non-negatively graded DG $P$ algebras. This is the result that allows us to transition between simplicial commutative algebras and commuatative DG algebras, thereby proving some results (see, e.g., ...