  We consider families of Abelian integrals arising from perturbations of
planar Hamiltonian systems. The tangential center focus problem asks for the
conditions under which these integrals vanish identically. The problem is
closely related to the monodromy problem, which asks when the monodromy of a
vanishing cycle generates the whole homology of the level curves of the
Hamiltonian. We solve both these questions for the case when the Hamiltonian is
hyperelliptic. As a side-product, we solve the corresponding problems for the
"0-dimensional Abelian integrals" defined by Gavrilov and Movasati.
