In this presentation we shall discuss construction, theoretical justification, and numerical testing of finite volume element discretizations of steady-state and transient convection-diffusion problems in 2 and 3 dimensions on unstructured grids. First, we derive locally conservative schemes on an arbitrary grid using finite volume method. Further, we discuss a priori error estimates in various norms for 2- and 3-dimensional problems. In 2-D we show that our estimates are sharp. Next we discuss an error control of finite volume discretizations. Using the ideas known from the finite element method we derive estimators based on local residuals and least-squares projections of the gradient for finite volume approximations and justify them theoretically. Namely, under appropriate assumptions, we prove that these estimates are reliable and efficient. This presentation is largely based on our joint work available on