The equations of transient heat conduction are developed for a generalized curvilinear coordinate system in Euclidean space. The equation is then specialized for the case of two-dimensional heat flow. The equations are applicable not only to planar problems but also to problems for thin surfaces. A compact finite-difference formulation for surface heat flow is given that permits variable mesh spacing, variable thickness, properties that vary spatially and with temperature, nonlinear boundary conditions, and volumetric heat generation. Example problems in nonorthogonal planar coordinates are given along with a problem for a thin curved surface.