We formulate and study the problem of varying G in Newtonian gravitation and cosmology. Exact solutions and all asymptotic cosmological behaviours are found for universes with G ∞t−n. Generalizations of Meshcherskii's theorem are proved for universes containing a perfect-fluid equation of state. The inequivalence of a positive cosmological constant and a p = — ρ stress is investigated when G varies and limits are placed on the rate of decrease of G (t) in order for de Sitter expansion to be attained as t → ∞. The conditions for more general forms of inflationary universes to occur are determined and new forms of inflation are found. These results are related to the more complicated systems of equations governing scalar—tensor gravity theories, their solutions and conformai properties.