The occurrence of large values for the sums S(χ, x) = Σn ≤xχ(n), where χ is a primitive character (mod q), is investigated. It is shown that the Pólya-Vinogradov bound O(√qlog q) for S(χ, x) is attained only very rarely, and a more precise bound that depends on rational approximations to is given. Moreover, improved values for the constant in the Pólya-Vinogradov inequality are obtained.