Quadratic forms of low dimension

I will show how the exceptional (or accidental) isomorphisms of low rank linear algebraic groups manifest themselves in beautiful classification results for quadratic forms of rank at most 6. By quadratic form, I could mean a homogeneous polynomial over a field in which case these results have a number theoretic flavor, or a vector bundle with a perfect orthogonal pairing over a manifold or algebraic variety in which case these results have a more geometric flavor. The most interesting applications arise when we are willing to allow a small amount of degeneration to our quadratic form, and in this context the classification results are still not yet achieved in dimensions 5 and 6.