A central result in extremal set theory is `the Erdos-Ko-Rado Theorem' (1961) which investigates the maximum size of families X of k-subsets in [n] such that two members in X intersect with at least t elements.\\
Two families X and Y of k-subsets in [n] are called `cross t-intersecting' if, for every members A in X and
B in Y, we have that A and B intersect with at least t elements. The cross t-intersecting version of the Erdos-Ko-Rado Theorem was conjectured but still open.\\
In this talk we verify the conjecture for all integers t>13 except finitely many n and k for each fixed t. Our proofs make use of a weight version of the problem and randomness. This is joint work with Peter Frankl, Norihide Tokushige, and Mark Siggers.