Renormalization in the theory of a quantized scalar field interacting with the classical Einstein gravitational field is discussed. The scalar field obeys the generalization of the Klein-Gordon equation which is conformally invariant in the limit of vanishing mass. A generalized Kasner metric corresponding to an anisotropic expansion of the universe is considered. Results obtained in collaboration with S.A. Fulling and B.L. Hu are described, which show explicitly how the infinities appearing in the expectation value of the energy-momentum tensor can be absorbed through renormalization of the cosmological constant and the coefficients of a quadratic tensor appearing in a slightly generalized form of the Einstein equation. There is also a finite renormalization of the gravitational constant.