We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.