The name quasi-interpolation is used for a large number and variety of methods. The first step in the tutorial will be to identify defining properties of quasi-interpolants and to differentiate them from interpolation methods. Then the foundations, which are the famous Strang-Fix properties and Schoenbergs work on splines, will be discussed first.

Details on the mathematical analysis of quasi-interpolation are investigated in three directions, namely on the basis (spline spaces, radial basis functions) from which the approximation is taken, on the form and computation of the quasi-interpolants (point evaluations, averages, least squares), and the mathematical properties (existence, locality, convergence questions, precision). Emphasis will be on the multivariate construction using radial basis functions and scattered data.

In the last part, examples of applications are presented, including data approximation, diagnostics and solution of PDEs.