A major obstacle for a broader adoption of the finite element method (FEM) in clinical biomedical applications is the generation of the model, frequently too slow for the times imposed by the clinical practice. The algorithms for automatic mesh generation have greatly improved, but their adoption by the biomedical community is still limited. The aim of this work is to review the principal algorithms for automatic mesh generation and to critically discuss them with particular reference to their applicability in the biomedical field. Specialized literature on numerical methods was reviewed in order to identify the main theoretical approaches currently available for automatic mesh generation. Then, published methods for the automatic generation of finite element models of organs from computed tomography data were reviewed and classified with a proposed taxonomy. Each method was reconnected to a theoretical approach described in the specialized literature whenever possible. Last, each method was critically reviewed with respect to its applicability to the clinical practice. None of the methods described satisfy all the requirements in terms of automation, generality, accuracy, and robustness imposed by a clinical application. However, some of these methods can already be successfully used in various application contexts, and a few guidelines are drawn.