To each nonzero sequence s := {s_n}_n≥0 of real numbers, we associate the Hankel determinants D_n = detH_n of the Hankel matrices H_n := (s_i+j)ⁿ _i,_j=0, n ≥ 0, and the nonempty set N_s := {n ≥ 1 D_n-1 ≠ 0}. We also define the Hankel determinant polynomials P₀ := 1, and P_n, n ≥ 1 as the determinant of the Hankel matrix H_n modified by replacing the last row by the monomials 1, x, xⁿ. Clearly Pn is a polynomial of degree at most n and of degree n if and only if n ε N_s. Kronecker established in 1881 that if N_s is finite then rank H_n = r for each n ≥ r - 1, where r := max N_s. By using an approach suggested by Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence {t_n}_n≥0 to be of the form t_n = D_n, n ≥ 0 for a real sequence {s_n}_n≥0. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial P_n satisfying deg P_n = n ≥ 1 is preceded by a nonzero polynomial P_n-1 whose degree can be strictly less than n - 1 and which has no common zeros with Pn. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that D_{0 > 0}, D_r-1 > 0 and D_n = 0 for all n ≥ r.