On the solubility of differential equations with simple characteristics

A survey is given of papers devoted to the problem of the existence of solutions to linear differential and pseudo-differential equations of principal type. The main results in this field are due to Lewy, Hцrmander, Nirenberg, Tréves and the author. We also give a new theorem of maximal generality on local solubility of equations of principal type. By way of illustration to the exposition we mention as examples: the Lewy operator; the operator arising from the solution of the problem with directional derivatives for elliptic second order equations; non-singular operators.