We present a categorification of the well-known Heisenberg algebra action on the cohomology of the moduli space of sheaves on surfaces (a classical result due to Grojnowski and Nakajima in the rank 1 case, and Baranovsky in higher rank). Time permitting, I will mention applications such as the AGT correspondence for general surfaces, or lifting Lehn's formula to the Chow groups of Hilbert schemes