T-splines are an important tool in IGA since they allow local refinement. In this paper we define analysis-suitable T-splines of arbitrary degree and prove fundamental properties: Linear independence of the blending functions and optimal approximation properties of the associated T-spline space. These are corollaries of our main result: A T-mesh is analysis-suitable if and only if it is dual-compatible. Indeed, dual compatibility is a concept already defined and used in L. Beirão da Veiga et al.5 Analysis-suitable T-splines are dual-compatible which allows for a straightforward construction of a dual basis. © 2013 World Scientific Publishing Company.