  We review theories of phosphorescence in cyclometalated complexes. We focus
primarily on pseudooctahedrally coordinated $t_{2g}^6$ metals (e.g.,
[Os(II)(bpy)$_3$]$^{2+}$, Ir(III)(ppy)$_3$ and Ir(III)(ptz)$_3$) as, for
reasons that are explored in detail, these show particularly strong
phosphorescence. We discuss both first principles approaches and semi-empirical
models, e.g., ligand field theory. We show that together these provide a clear
understanding of the photophysics and in particular the lowest energy triplet
excitation, T$_1$. In order to build a good model relativistic effects need to
be included. The role of spin-orbit coupling is well-known, but scalar
relativistic effects are also large - and are therefore also introduced and
discussed. No expertise in special relativity or relativistic quantum mechanics
is assumed and a pedagogical introduction to these subjects is given. It is
shown that, once both scalar relativistic effects and spin-orbit coupling are
included, time dependent density functional theory (TDDFT) provides
quantitatively accurate predictions of the radiative decay rates of the
substates of T$_1$ in phosphorescent organotranstion-metal complexes. We
describe the pseudo-angular momentum model, and show that it reproduces the key
experimental findings. For example, this model provides a simple explanation of
the relative radiative rates of the substates of T$_1$, which differ by orders
of magnitude. Special emphasis is placed on materials with potential
applications as active materials in organic light-emitting diodes (OLEDs) and
principles for the design of new complexes are identified on the basis of the
insights provided by the theories reviewed. We discuss the remaining
theoretical challenges, which include deepening our understanding of solvent
effects and, vitally, understanding and predicting non-radiative decay rates.
