Given a compact subset, E in R, the Chebyshev polynomials are the unique minmizers of the sup norm over E among all degree n monic polynomials. I will describe some recent results of Christensen, Simon and inchenko on this classical subject. We settle a 45 year old conjecture of Widom on the large n pointwise asymptotics for the case of finite gap sets. We also extend upper bounds on the norm that Totik and Widom obtained for the finite gap case to positive measure Cantor sets. Our proof in this case is new and even in the finite gap case is simpler and more explicit bounds than the earlier work.