We study complex spatial quartic surfaces with simple singularities and give their classication up to equisingular deformation. Simple quartics are K3-surfaces and as such they can be studied by means of the global Torelli theorem and the surjectivity of the period map combined with Nikulin's theory of discriminant forms. We reduce the classification problem to a certain arithmetical problem concerning lattice extensions. Then, based on Nikulin's existence criterion, we list all sets of simple singularities realized by non-special quartics; the result is stated in terms of perturbations of a few extremal sets. For each realizable set of singularities, we use Miranda{Morrison's theory to give a complete description of the connected components of the corresponding equisingular stratum.