We analyze the approximation properties of the k-method. The k-method is a finite element technique where spline basis functions of higher-order continuity are employed. It is a fundamental feature of the new field of isogeometric analysis. It has been shown that using the k-method has many advantages over the classical finite element method in application areas such as structural dynamics, wave propagation, and turbulence. In this talk, we first investigate the approximation properties of the k-method utilizing the notions of Kolmogorov n-width, sup-inf, and approximation ratio. These three tools were introduced in order to assess the effectiveness of approximating functions. Following a review of theoretical results, we present the results of a numerical study in which the n-width and sup-inf are computed for a number of one-dimensional, multi-dimensional, and rational cases. This study sheds further light on the approximation properties of the k-method. In addition, we present a comparison study of the k-method and the classical finite element method. We conclude this talk with a discussion of local approximation behavior around singularities and the benefits of reducing continuity near selected features such as cusps and boundary layers.