A framework to study quantum field theory on the de Sitter and anti-de Sitter manifolds will be described. In this context, we study particle decay in de Sitter space as given by first order perturbation theory in an interacting quantum field theory. We show that for fields with masses above a critical mass $m_c$ there is no such thing as particle stability, so that decays forbidden in flat space-time do occur there. The lifetime of such a particle also turns out to be independent of its velocity when that lifetime is comparable with de Sitter radius. Particles with lower mass are even stranger: their decay can only yield quantified masses, with zero lifetime.