Abstract: In this talk, we will discuss the cohomology groups of the free loop space of a topological space $X$, which is the space of all continuous maps from $S^1$ to $X$. Using the Eilenberg–Moore spectral sequence, we will connect these cohomology groups and Hochschild homology and compute them in specific cases. We will mostly follow the article "On the Characteristic Zero Cohomology of the Free Loop Space" by L. Smith.