In the framework of the standard model of cosmology and particle physics (the so-called concordance model) the observed accelerated expansion of the Universe today is driven by a substance of unknown nature with a negative pressure, the so-called cosmological constant. The requirement for the existence of a cosmological constant (or any dynamical dark energy) follows from the Einstein equations, which assume general relativity (GR) as a true theory of gravity, and massless of the graviton, the helicity-2 particles associated with gravitational radiation. Despite the success of the concordance model, there are several unanswered questions that motivate us to consider as an alternative the possibility that GR is not suitable to describe physical processes at cosmological scales. One of the most active areas of research in gravity theory stems from the assumption to give the graviton a non-zero mass (massive gravity). However, to formulate massive gravity is a challenging task and was addressed since 1930. More groundbreaking progress was made through the formulation of the ghost-free massive gravity deRham-Gabadadze-Tolley (dRGT 2010) theory, and its bigravity generalization (Hassan and Rosen 2011). In my talk I will review possibilities to constrain the graviton mass through current and future observations. The focus of my talk will be pulsar timing array (PTA) data viewed in the context of massive gravity.