Since their introduction in 1972 by Clutton-Brock, so-called biorthogonal basis sets have become a popular tool in galactic dynamics. They provide a method of solving Poisson's equation that scales linearly with the number of particles in a galaxy or halo simulation, and have also received wide attention in the perturbation theory of disk galaxies.

This thesis begins by discussing the theory behind such basis sets, and the context of their applications. Some introductory results are presented, deriving the possible functional forms that basis sets can take, and identifying the lack of suitably flexible basis sets in the literature as a stumbling block to further usage of the technique.

Subsequently, several new families of basis sets are derived, whose free parameters and resemblance to classic double-power law formulas provide a much needed increase in flexibility for the modelling of realistic galaxies and haloes. Along the way a simple yet under studied spherical double-power model is described that interpolates between the properties of the more famous NFW and Hernquist models.

Finally, we turn to an application of the new basis sets: the problem of efficiently re-simulating cosmological dark matter haloes. The ability to place new objects in a realistic time-evolving gravitational potential is desirable from a modelling point of view, as it permits us to constrain the properties of massive, unseen galactic components (dark haloes) via visible dynamical tracers such as stellar streams.

The thesis concludes by suggesting some further practical applications of the basis set technique in astronomy, as well as some pointers towards future theoretical developments and applications to other areas of physics.