In bodies where discontinuities, like cracks, emerge and interact, the classical equations for heat and mass transfer are not well suited. We propose a peridynamic model for transient heat (or mass) transfer which is valid when the body undergoes damage or evolving cracks. We use a constructive approach to find the peridynamic formulation for heat transfer and test the numerical convergence to the classical solutions in the limit of the horizon (the nonlocal parameter) going to zero for several one-dimensional problems with different types of boundary conditions. We observe an interesting property of the peridynamic solution: when two m-convergence curves, corresponding to two different horizons, for the solution at a point and an instant intersect, the intersection point is also the exact classical (local) solution. The present formulation can be easily extended to higher dimensions and be coupled with the mechanical peridynamic description for thermomechanical analyses of fracturing bodies, or for heat and mass transfer in bodies with evolving material discontinuities. (C) 2010 Elsevier Ltd. All rights reserved.