Additive manufacturing, is a revolutionary development of manufacturing playing an ever increasing role in our everyday lives. It has many applications in the precise manufacturing in various industries such as mechanical, aerospace, and medical.From a mechanics point of view, understanding and being able to predict and control the residual stresses is crucial in order to tailor and design the process in such a way that the manufactured piece meets the required properties in its working conditions.Indeed, the high temperatures and natural cooling to which the piece is subjected to during the additive manufacturing process causes large strains and can result in a high level of residual stresses.This may lead to severe part distortion, dimensional inaccuracies, and even cracks in the final manufactured object. Such conditions put the additive manufacturing problem beyond the scope of the application of linearized elasticity.However, to the best of our knowledge, most of the existing works on additive manufacturing in the literature are purely computational, are based on linearized elasticity, and use finite element analysis. It should be emphasized that the boundary-value problems of accretion problems have not been formulated properly to this date and the mechanics of accretion at finite strains is still at its infancy. Additive manufacturing advances are good motivations for building a mathematical theory of the mechanics of accretion at finite strains that can be used in these new technologies.Our goal in the proposed research program is to build a general model of thermoelastic surface growth where surface growth, i.e., the act of adding material to the boundary of the workpiece, is coupled with the machine-induced heating and natural cooling during the solidification process of the added material.For such structures, a knowledge of the mechanical properties of the constituents in their stress-free configuration is not sufficient. The distribution of residual stresses explicitly depends on the accretion process, e.g., initial temperature of the added material at the time of attachment, accretion velocity, history of loading, etc.In previous works we introduced a geometric theory of nonlinear accretion mechanics for symmetric surface growth of cylindrical and spherical bodies. We used this theory for the analysis of several model problems. In the proposed research program, the theory of our previous works will be extended to consider thermal effects and the assumption of symmetry will be released. These generalizations are quite nontrivial and challenging. Several numerical examples will be considered in this investigation. Studied cases will concern both isotropic and anisotropic materials.