Let {B n (x)} denote Bernoulli polynomials. In this paper we generalize Kummer's congruences by determining B k ( p - 1 ) + b (x)/(k(p-1)+b)(modp n ), where p is an odd prime, x is a p-integral rational number and p-1 b. As applications we obtain explicit formulae for x = 1 p - 1 (1/x k )(modp 3 ), x = 1 ( p - 1 ) / 2 (1/x k )(modp 3 ),(p-1)!(m odp 3 ) and A r (m,p)(modp), where k {1,2,...,p-1} and A r (m,p) is the least positive solution of the congruence px=r(modm). We also establish similar congruences for generalized Bernoulli numbers {B n , χ }.