Stable minimal hypersurfaces are a central topic in both Riemannian geometry and geometric analysis. Starting with the foundational work of Fischer-Colbrie and Schoen, the last decades have seen an intense research activity aimed to understand the geometry and topology of this important class of submanifolds. In this talk, I will report about some recent results concerning vanishing theorems for L2 harmonic forms and L2 harmonic spinors on stable minimal hypersurfaces. The talk is based on a joint work with Giuseppe Pipoli