  In this paper rational solutions of the fifth Painlev\'e equation are
discussed. There are two classes of rational solutions of the fifth Painlev\'e
equation, one expressed in terms of the generalised Laguerre polynomials, which
are the main subject of this paper, and the other in terms of the generalised
Umemura polynomials. Both the generalised Laguerre polynomials and the
generalised Umemura polynomials can be expressed as Wronskians of Laguerre
polynomials specified in terms of specific families of partitions. The
properties of the generalised Laguerre polynomials are determined and various
differential-difference and discrete equations found. The rational solutions of
the fifth Painlev\'e equation, the associated $\sigma$-equation and the
symmetric fifth Painlev\'e system are expressed in terms of generalised
Laguerre polynomials. Non-uniqueness of the solutions in special cases is
established and some applications are considered. In the second part of the
paper, the structure of the roots of the polynomials are investigated for all
values of the parameter. Interesting transitions between root structures
through coalescences at the origin are discovered, with the allowed behaviours
controlled by hook data associated with the partition. The discriminants of the
generalised Laguerre polynomials are found and also shown to be expressible in
terms of partition data. Explicit expressions for the coefficients of a general
Wronskian Laguerre polynomial defined in terms of a single partition are given.
