Any theorem on the existence and uniqueness of a system of nonlinear equations?

Question

I'm now solving a system of nonlinear equations (not differential equations). Although I'm able to find a solution for some parameter sets, there are also cases when I can't find a solution. So I'm quite curious about the existence and uniqueness of a system of nonlinear equations. In a lecture note I found it's said a system of $N$ equations with $N$ unknowns in general has a unique solution. But it's just a brief statement, without a reference or proof. So I wonder if there's any such a theorem.