The Prime Number Theorem tells us that the logarithmic integral, li(x)’>li(x)li(x) , is a good approximation to π(x)’>π(x)π(x) , the number of primes up to x. Numerically it always seems to be an overestimate, so π(x)−li(x)’>π(x)−li(x)π(x)−li(x) is negative. The first point where this ceases to be the case is known as Skewes’ number whose true value is as yet unknown. I will report on joint work with Chris Smith and Dave Platt, where we improve the best upper bound on Skewes’ number.