Schrodinger's equation has been known for more than 90 years, yet many pressing questions in electronic structure theory remain unanswered. Quantum Chemistry is a successful field: the weak correlation problem has been solved; we can get "the right answer for the right reason" at reasonably low polynomial computational cost instead of the combinatorial expense of brute force approaches. Despite many advances, however, the so-called strong correlation problem, where the chemist's molecular orbital picture breaks down, remains quite open, particularly for large molecules and solids. This seminar will address challenges and progress on the treatment of strong correlation, and its connection with quantum symmetries and degeneracy. Methods developed in our research group for strong correlation will be discussed, and applications to fullerenes and other carbon nanostructures will be presented.