In recent years, many composite steel-concrete cable-stayed bridges with medium and long spans have been designed and constructed. In the last two decades, extensive research has been carried out on the structural analysis of cable-stayed bridges, several commercial software packages being available today. However, few analysis results of composite cable-stayed bridges have been published, in particular presenting the nonlinear analysis from construction stages up to the structural failure. This paper presents the nonlinear structural analysis of composite steel-concrete cable-stayed bridges. The composite deck and the concrete towers are modelled by three node / 7dof steel and concrete frame elements, stays and shear connection of the deck are modelled using cable and continuous spring elements. Geometrical and material nonlinear behaviours of both steel and concrete materials are considered. Time dependent effects due to load history, creep, shrinkage and aging of concrete are also included in the analysis. An evaluation of the main sources of non-linearity is made for the case of a 420 m main-span composite cable-stayed bridge, at service conditions. The failure load and failure mechanism of this composite cable-stayed bridge is also investigated, both at the end of construction and at long-term, being analysed the influence of several factors, namely: 1) the deck load pattern, 2) the time dependent effects, 3) the intermediate piers in the lateral spans, 4) the effective slab width, and 5) the flexible connection.