Does anyone know the classification of fourth order surfaces? By "fourth order surface" I mean a surface defined by an equation of the form $$f(x, \, y, \, z)=0,$$ where $f$ is a polynomial of degree $4$.

**Edit.** The two answers below (J. Silverman and F. Polizzi) concern the case of complex surfaces. Are there known results in the case of *real* surfaces?