A classical theorem of Joyal and Street establishes an equivalence between braided categorical groups and quadratic forms. This brings an important geometric insight into the theory of braided fusion categories: one can treat them as non-commutative geometric objects. From this point of view the Drinfeld centers correspond to hyperbolic quadratic forms. We use this observation to define a categorical analogue of the classical Witt group of quadratic forms. It turns out that the categorical Witt group W is no longer a torsion group. We discuss the structure of W and its generalizations: the super and equivariant categorical Witt groups. This talk is based on joint works with Alexei Davydov, Victor Ostrik, and Michael Mueger.