This chapter covers the basics of cubature rules. Starting from the definition of polynomial spaces and cubature rules, it discusses interpolatory cubature rules, Tchakaloff’s theorem on positive cubature rules, and Sobolev’s theorem on invariant cubature rules. It provides product-type cubature rules on several regular domains, such as product domains, balls, and simplexes, as well as invariant cubature rules on these domains, and a brief section on constructing cubature rules numerically.

The chapter contains an extensive family of minimal or near-minimal cubature rules on the square and an unbounded domain. Those on the square extended minimal cubature rules for the Chebyshev weight functions in several aspects.

The chapter explains another angle of looking at minimal cubature rules, using the language of ideal and variety in algebraic geometry. In essence, the existence of a cubature rule of degree m amounts to the existence of a polynomial ideal generated by m-orthogonal polynomials with zero-dimensional and real variety, and the codimension of the ideal equals the cardinality of the variety. The abstract point of view pinpoints the root of the difficulty in understanding the minimal cubature rules and indicates a theoretical roadmap.

Many questions around cubature rules remain open. The chapter discusses two open problems; both are fundamental for further study. The first is about better lower bounds of the number of nodes, and the second discusses cubature rules of more than two variables.