Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Federico Cantero Morán[en] We will compute the cohomology groups of the unordered configuration spaces of the plane $\mathbb{R}^2$ with $\mathbb{Z}$ coefficients, and with $\mathbb{Z}_p$ we’ll only give the proof in case $p = 2$. We will use a cell complex by Fuks and Vainshtein.