The Betti numbers of a graded module over the polynomial ring form a
table of numerical invariants that refines the Hilbert polynomial. A
sequence of papers sparked by conjectures of Boij and S\"oderberg have led to the
characterization of the possible Betti tables up to rational multiples~Wthat is, to the rational
cone generated by the Betti tables. I will summarize this work by describing the cone and the
closely related cone of cohomology tables of vector bundles on projective space, and
I will explain some of the applications of the theory, including the one that
originally motivated the conjectures of Boij and S\"oderberg, a proof of the Multiplicity
Conjecture of Herzog, Huneke and Srinivasan.