Trefftz-type elements, or T-elements, are finite elements the internal field of which fulfils the governing differential equations of the problema priori whereas the interelement continuity and the boundary conditions are enforced in an integral weighted residual sense or pointwise. Although the key ideas of such elements can be traced back to Jirousek and Leon in 1977, the T-element approach has received serious attention only for the past ten years. The T-element approach makes it possible to generate highly accurate h- or p-elements exhibiting many important advantages over their more conventional counterpart. The paper surveys existing T-element formulations (including some yet unpublished ones) and assesses critically their performance (accuracy, h- and p-convergence, sensitivity to mesh distortions, handling of singularities and geometry or load induced local effects, etc.). The available applications include plane elasticity, thin and thick plates, cylindrical shells, axisymmetric 3-D elasticity, Poisson's equation and transient heat conduction analysis. Existing approaches to adaptive reliability assurance based on p-extension are also discussed and future trends in the T-element research shortly outlined.