### ABSTRACT

The earliest reference to numerical integration over three
dimensional regions which I can find is 1877, in a paper familiarly
referred to as "Maxwell's Brick". Since then the topic of numerical
integration has been the subject of probably thousands of papers. Given
a sufficiently smooth function *f* from R^{s} to R,
the Cubature problem
is to find numerical approximations to integrals over regions in
s-dimensional space. This talk will describe a general class of methods
to approximate such integrals. The methods are characterized by structure
parameters which are chosen to satisfy certain constraints, while
minimizing the number of function evaluations required. Some
simple examples will be given.