  A bilinear quadrature numerically evaluates a continuous bilinear map, such
as the $L^2$ inner product, on continuous $f$ and $g$ belonging to known
finite-dimensional function spaces. Such maps arise in Galerkin methods for
differential and integral equations. The construction of bilinear quadratures
over arbitrary domains in $\mathbb{R}^d$ is presented. In one dimension,
integration rules of this type include Gaussian quadrature for polynomials and
the trapezoidal rule for trigonometric polynomials as special cases. A
numerical procedure for constructing bilinear quadratures is developed and
validated.
