  Let $S_{0}, S_{1}$ and $S_{2}$ be connected Riemann surfaces and let
$\beta_{1}:S_{1} \to S_{0}$ and $\beta_{2}:S_{2} \to S_{0}$ be surjective
holomorphic maps. The associated fiber product
  $S_{1} \times_{(\beta_{1},\beta_{2})} S_{2}$ has the structure of a singular
Riemann surface, endowed with a canonical map $\beta$ to $S_{0}$ satisfying
that $\beta_{j} \circ \pi_{j}=\beta$, where $\pi_{j}$ is coordinate projection
onto $S_{j}$. In this paper we provide a Fuchsian description of the fiber
product and obtain that if one the maps $\beta_{j}$ is a regular branched
cover, then all its irreducible components are isomorphic.
  In the case that both $\beta_{j}$ are of finite degree, we observe that the
number of irreducible components is bounded above by the greatest common
divisor of the two degrees; we study the irreducibility of the fiber product.
  In the case that $S_{0}=\widehat{\mathbb C}$, and $S_{1}$ and $S_{2}$ are
compact, we define the strong field of moduli of the pair $(S_{1}
\times_{(\beta_{1},\beta_{2})} S_{2},\beta)$ and observe that this field
coincides with the minimal field containing the fields of moduli of both pairs
$(S_{1},\beta_{1})$ and $(S_{2},\beta_{2})$. Finally, in the case that the
fiber product is a connected Riemann surface, we provide an isogenous
decomposition of its Jacobian variety.
