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We introduce the nonlocal Cattaneo-Vernotte equation by including a relaxation effect in the nonlocal diffusion equation, both of which are models for anomalous diffusion. This equation has two different interpretations: as a generalization of Fick’s first law in terms of a nonlocal flux and memory kernel and as an equation arising from the generalized master equation for a continuous time random walk. Both interpretations are discussed and the latter describes a scaling of relaxation time and nonlocality. A relationship to fractional diffusion equations in the limit of vanishing relaxation time and nonlocality is also established. The main contribution of this paper is to introduce nonlocal boundary conditions for the nonlocal Cattaneo-Vernotte equation, and the ensuing variational and finite element formulations. Examples are given where the effects of relaxation time and nonlocality are studied.