  We present a five-step method for the calculation of eigenvalue correlation
functions for various ensembles of real random matrices, based upon the method
of (skew-) orthogonal polynomials. This scheme systematises existing methods
and also involves some new techniques. The ensembles considered are: the
Gaussian Orthogonal Ensemble (GOE), the real Ginibre ensemble, ensembles of
partially symmetric real Gaussian matrices, the real spherical ensemble (of
matrices $A^{-1}B$), and the real anti-spherical ensemble (consisting of
truncated real orthogonal matrices). Particular emphasis is paid to the
variations in method required to treat odd-sized matrices. Various universality
results are discussed, and a conjecture for an `anti-spherical law' is
presented.
