This is the first of a two-part contribution devoted to the evolution of a massless scalar field in various black-hole spacetimes; the second part follows this one and is authored by William G. Laarakkers. In this contribution we consider the evolution of a scalar field propagating in Schwarzschild-de Sitter spacetime. The field is non-minimally coupled to curvature through a coupling constant &xgr;. The spacetime possesses a cosmological horizon in addition to the usual event horizon. The presence of this new horizon affects the late-time evolution of the scalar field: The usual inverse power-law decay of pure Schwarzschild spacetime is replaced by a faster, exponential decay. The decay constant is proportional to the surface gravity of the cosmological horizon, and depends on the field’s multipole orderland the constant &xgr;. ©1999 American Institute of Physics.