There has been very convincing numerical evidence since the 1970s that the positions of zeros of the Riemann zeta function and other L-functions show the same statistical distribution (in the appropriate limit) as eigenvalues of random matrices.  Proving this connection, even in restricted cases, is difficult, but if one accepts the connection then random matrix theory can provide unique insight into long-standing questions in number theory.  I will give a history of the attempt to prove the connection, as well as propose that the way forward may be to forgo the enticing beauty of the determinantal formulae available in random matrix theory in favour of something a little less elegant.