Hermitian K-theory is an invariant that can be defined for every scheme X.
Traditionally the focus has been on schemes X where 2 is invertible. Recently
an understanding has emerged of how to deal with the many different variants
that are available when 2 is not invertible. In this talk I will survey the
various known and unknown but expected properties of the hermitian K-theory
presheaf, and their computational consequences. The results in this talk are
joint work with a many people, among which are B. Calmès, E. Elmanto, Y. Harpaz,
J. Shah and L. Yang.