  We develop a renormalisation scheme for time--ordered products in interacting
field theories on curved spacetimes which consists of an analytic
regularisation of Feynman amplitudes and a minimal subtraction of the resulting
pole parts. This scheme is directly applicable to spacetimes with Lorentzian
signature, manifestly generally covariant, invariant under any spacetime
isometries present and constructed to all orders in perturbation theory.
Moreover, the scheme captures correctly the non--geometric state--dependent
contribution of Feynman amplitudes and it is well--suited for practical
computations. To illustrate this last point, we compute explicit examples on a
generic curved spacetime, and demonstrate how momentum space computations in
cosmological spacetimes can be performed in our scheme. In this work, we
discuss only scalar fields in four spacetime dimensions, but we argue that the
renormalisation scheme can be directly generalised to other spacetime
dimensions and field theories with higher spin, as well as to theories with
local gauge invariance.
