  Let $X$ be a compact Riemann surface of genus $g$. Jacobi's inversion theorem
states that the Abel-Jacobi map $\varphi : X^{(g)} \longrightarrow J(X)$ is
surjective, where $X^{(g)}$ is the symmetric product of $X$ of degree $g$ and
$J(X)$ is the Jacobi variety of $X$. Riemann obtained the explicit solution of
the Jacobi inversion problem introducing Riemann's theta functions. We study
such a problem for singular curves. We define a generalization of Riemann's
theta functions and Riemann's constants. We obtain similar results for singular
curves.
