In the last few years there has been a great amount of interest in Random Constraint Satisfaction Problems, both from an experimental and a theoretical point of view. Quite intriguingly, experimental results with various models for generating random CSP instances suggest that the probability of such problems having a solution exhibits a ?threshold-like? behavior. In this spirit, some preliminary theoretical work has been done in analyzing these models asymptotically, i.e., as the number of variables grows. In this paper we prove that, contrary to beliefs based on experimental evidence, the models commonly used for generating random CSP instances do not have an asymptotic threshold. In particular, we prove that asymptotically almost all instances they generate are overconstrained, suffering from trivial, local inconsistencies. To complement this result we present an alternative, single-parameter model for generating random CSP instances and prove that, unlike current models, it exhibits non-trivial asymptotic behavior. Moreover, for this new model we derive explicit bounds for the narrow region within which the probability of having a solution changes dramatically