The cochains of a topological space X naturally form an associative (non-commutative) algebra under the cup product, so they give rise to Bar and Hochschild complexes. These complexes play a central role in algebra, representation theory, and mathematical physics, but what does their cohomology measure about the topology of X?

After reviewing the construction of these two complexes, I'll discuss old and new results about the topological invariants they afford: they measure loop spaces of X. In particular, we will obtain new ways to measure elements in the fundamental group of X.