  In this paper we study the orthogonality conditions satisfied by Jacobi
polynomials $P_n^{(\alpha,\beta)}$ when the parameters $\alpha$ and $\beta$ are
not necessarily $>-1$. We establish orthogonality on a generic closed contour
on a Riemann surface. Depending on the parameters, this leads to either full
orthogonality conditions on a single contour in the plane, or to multiple
orthogonality conditions on a number of contours in the plane. In all cases we
show that the orthogonality conditions characterize the Jacobi polynomial
$P_n^{(\alpha, \beta)}$ of degree $n$ up to a constant factor.
