The properties of surjective real quadratic maps are investigated.Sufficient conditions for the property of surjectivity to be stable under various perturbations are obtained. Examples of surjective quadratic maps whose surjectivity breaks down after an arbitrarily small perturbation are constructed. Sufficient conditions for quadratic maps to have nontrivial zeros are obtained. For a smooth even map in a neighbourhood of the origin an inverse function theorem in terms of the degree of the orresponding quadratic map is obtained. A canonical form of surjective quadratic maps from R^3 to R^3 is constructed.