Let : Xg -> X_ g0 be a m-sheeted covering of Riemann surfaces of genus g and g0. The goal is to nd properties that Xg (or rather, the Jacobian of Xg) has, due to the existence of the covering . This is an old problem that goes back to Riemann and Jacobi which is solved via the theta functions of the Xg. Many other mathematicians have worked on the cases of small genus and small degree, most notably Frobenius, Prym, Konigsberger, Rosenhein, Gopel, among others. The main goal of this talk is to discuss determining relations among theta-nulls in the non-hyperelliptic case for g > 2. The vanishing theta-nulls for hyperelliptic curves were studied by 19-th century mathematicians and are well understood.