It is a classical fact that for closed manifolds \(X\) the homotopy classes of maps \(X^n\rightarrow S^n\) are classified by their degree. The Pontryagin-Thom construction provides a similar construction when \(X\) and the sphere have also different dimensions, and thus generalizes the notion of degree. In particular, the homotopy classes of maps \(X^{n+1}\rightarrow S^n\) are in one-to-one correspondence with framed circles up to framed cobordism in \(X\), and the corresponding set comes equipped with a group structure. In this talk, we introduce the Pontryagin-Thom construction and the concept of framed cobordism classes, and we compute the group of homotopy classes of maps \(X^{n+1}\rightarrow S^n\) in terms of geometric and topological information of \(X\). If time permits, we delve into some ideas of the proof, and discuss applications to vector bundles.