The basic idea of approximate approximations is simple: we take approximations to functions where the approximants depend on parameters such as a step size. Classically one wishes to establish convergence theorems as the step size goes to zero. Also, we are usually interested in an order of convergence or an approximation order. However, in many cases, although good approximations can be made, convergence to zero is ultimately not achieved. Nonetheless, the approximations are highly useful if the remaining error can be singled out and kept below machine precision for example. This leads to important mathematical applications, such as the approximation of integral operators and cubature formulas. A variety of important generalizations and applications can be addressed, e.g. approximation with wavelets, general grids and scattered data, algorithms for solving differential and integral equations, and boundary value problems. Altogether the approximate approximations proved to be a useful alternative to the classical ideas of strict convergence. This Chapter starts with simple examples of second- and high-order quasi-interpolants in both one-dimensional and multi-dimensional cases. Then, in the significant case of the harmonic potential, we describe a fast method of arbitrarily high order for approximating volume potentials. Our method is based on the use of approximate quasi-interpolants and separated representations and is also effective in high-dimensional cases. Details and proofs are given in Chaps. 2 and 3 .