TITLE: Does anyone know the classification of fourth order surfaces?
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: The classification of real non-singular quartic surfaces in $\mathbb{RP}^3$ (up to various possible equivalence relations like homeomorphism, isotopy,...) is discussed in 

A. Degtyarev, V. Kharlamov: Topological properties of real algebraic varieties: Rokhlin's way. Uspekhi Mat. Nauk 55 (2000), no. 4(334), 129-212; translation in
Russian Math. Surveys 55 (2000), no. 4, 735–814

See also Sections 3.5.3 and 3.5.4 in the arXiv-version of the paper (as well as the references given there).
Some statements concerning the structure and classification of singular real quartic surfaces are given in:

A. Degtyarev and I. Itenberg: On real determinantal quartics. Proceedings of the Gökova Geometry-Topology Conference 2010, 110–128, Int. Press, Somerville, MA, 2011.