In the 30s, Witt initiated the study of quadratic forms over a general field by introducing the Witt group of a field. This study gained great momentum in the 60s by Pfisters theory of multiplicative forms, now called Pfister forms. Pfisters theory led to the solution of several longstanding problems on quadratic forms and several new quesions and conjectures were thrown open. One such is the Milnor conjecture on the classification of quadratic forms via Galois cohomological invariants. The first few of these invariants, the so-called classical invariants, can be explained in terms of the Clifford algebra of the quadratic form. In this talk, we shall describe some major developments in this direction.