  We review the theory of orthogonal separation of variables of the
Hamilton-Jacobi equation on spaces of constant curvature, highlighting key
contributions to the theory by Benenti. This theory revolves around a special
type of conformal Killing tensor, hereafter called a concircular tensor. First,
we show how to extend original results given by Benenti to intrinsically
characterize all (orthogonal) separable coordinates in spaces of constant
curvature using concircular tensors. This results in the construction of a
special class of separable coordinates known as Kalnins-Eisenhart-Miller
coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation
algorithm, which uses concircular tensors to intrinsically search for
Kalnins-Eisenhart-Miller coordinates which separate a given natural
Hamilton-Jacobi equation. As a new application of the theory, we show how to
obtain the separable coordinate systems in the two dimensional spaces of
constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the
Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability
of the three dimensional Calogero-Moser and Morosi-Tondo systems.
