This thesis is an examination of the uses of mean-field theory in problems of the theory of strongly-correlated electronic systems, particularly to the problem of orbital ordering in transition metal oxides. We will apply mean-field theory to various models for orbital ordering of transition metal oxides, and also show that mean-field theory is not as bad an approximation as it might initially seem. We are also interested modelling superconductivity in heavy fermion systems. We conclude from our modelling on transition metal oxides that the mean-field theory we use based on the Stoner criterion will not be adequate to model such complicated phenomena. We propose an alternate mean-field theory based on non-linear fermionic transformations which we introduce. We suggest further improvements in the form of a non-orthogonal transformation, which we also introduce. As a diversion, we model frustrated antiferromagnetism on a pyrochlore lattice. The particular material is Gd\(_2\)Ti\(_2\)O\(_7\). We show that there are many effects in competition with each other. We conclude with a proposed magnetic structure which appears to be a better fit to experimental data than previous suggestions.