  This is an expos\'e on the use of O'Sullivan penalised splines in
contemporary semiparametric regression, including mixed model and Bayesian
formulations. O'Sullivan penalised splines are similar to P-splines, but have
an advantage of being a direct generalisation of smoothing splines. Exact
expressions for the O'Sullivan penalty matrix are obtained. Comparisons between
the two reveals that O'Sullivan penalised splines more closely mimic the
natural boundary behaviour of smoothing splines. Implementation in modern
computing environments such as Matlab, R and BUGS is discussed.
