This paper reviews the use of sufficient and ancillary statistics in c
onstructing conditional distributions for inference about a parameter.
Special emphasis is given to recent developments in accurate approxim
ation of densities, distribution functions and likelihood functions, a
nd to the role of conditioning in these approximations. Exact conditio
nal or marginal inference is available for essentially two classes of
models, exponential family models and transformation family models. Th
e approximations are very useful for practical implementation of these
exact results. The form of the approximations suggests methods for in
ference in more general families.