  The concept of weights on the cohomology of algebraic varieties was initiated
by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply
connected with the concept of motives and appeared first on the singular
cohomology as the weights of (possibly mixed) Hodge structures and on the etale
cohomology as the weights of eigenvalues of Frobenius. But weights also appear
on algebraic fundamental groups and in p-adic Hodge theory, where they become
only visible after applying the comparison functors of Fontaine. After
rehearsing various versions of weights, we explain some more recent
applications of weights, e.g., to Hasse principles and the computation of
motivic cohomology, and discuss some open questions.
