On the solvability and hypoellipticity of complex vector fields

The article is a survey of recent results about smooth complex vector fields with critical points (i. e., that vanish at certain points) from the viewpoint of hypoellipticity and local solvability. The results are rudimentary, as little is known on the subject so far, outside some relatively simple special cases. Several open problems are listed (Section 6). A stable class is isolated: the vector fields of principal type, as well as a subclass (in two dimensions), introduced under the name of quasi-elliptic vector fields (Section 5).