The complex space forms admit a particularly nice set of standard examples, catalogued by R. Takagi for CPn and by S. Montiel for CHn. The Takagi/Montiel list may be described as the set of complete Hopf hypersurfaces with constant principal curvatures. There is a vast literature characterizing various subsubsets of this list in terms of properties of the shape operator, Ricci tensor, etc. There are also significant properties enjoyed by these hypersurfaces (for example, Hopf, homogeneous, isoparametric, constant principal curvatures), that are shared with examples not on the Takagi/Montiel list. Much recent and current research has been devoted to describing and classifying such hypersurfaces.
I will begin with an introduction to the complex space forms CPn and CHn and the theory of their submanifolds, including the geometry of tubes and focal sets. I will then describe the hypersurfaces in the Takagi/Montiel list. Finally, I will discuss results about the wider class of hypersurfaces mentioned above.