In the paper, quadratic mappings acting from one finite-dimensional space to another are studied. Sufficient conditions for the stable surjectivity of a quadratic surjective mapping (i.e., for the condition that every quadratic mapping sufficiently close to a given one is also surjective) are obtained. The existence problem for nontrivial zeros of a surjective quadratic mapping acting from R^n to R^n is studied. For n = 3, the absence of these zeros is proved.