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This paper is concerned with solution algorithms for general convex vector optimization problems (CVOPs). So far, solution concepts and approximation algorithms for solving CVOPs exist only for bounded problems [Ararat et al. 2022, Doerfler et al. 2021, Loehne et al. 2014]. They provide a polyhedral inner and outer approximation of the upper image that have a Hausdorff distance of at most $\varepsilon$. However, it is well known (see [Ulus, 2018]), that for some unbounded problems such polyhedral approximations do not exist. In this paper, we will propose a generalized solution concept, called an $(\varepsilon,δ)$--solution, that allows also to consider unbounded CVOPs. It is based on additionally bounding the recession cones of the inner and outer polyhedral approximations of the upper image in a meaningful way. An algorithm is proposed that computes such $δ$--outer and $δ$--inner approximations of the recession cone of the upper image. In combination with the results of [Loehne et al. 2014] this provides a primal and a dual algorithm that allow to compute $(\varepsilon,δ)$--solutions of (potentially unbounded) CVOPs. Numerical examples are provided.