We study the Cauchy problem in the hyperbolic space ${ℍ}^n(n\ge 2)$ for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space ${ℝ}^n$ new phenomena arise, which depend on the properties of the diffusion process in ${ℍ}^n$. We also investigate a family of travelling wave solutions, named...