Abstract：

Non-singular Fano 3-folds were classified by Fano, Iskovskikh, Mori and Mukai, and are known to fall into 107 deformation families that are fairly well studied. The Mori category of projective 3-folds with terminal singularities is considerably more diverse. For many purposes, it is enough to concentrate on varieties with terminal orbifold points 1/r(1,a,r-a) as the only singular points. There are approximately 60,000 possible Hilbert series for these varieties, of which several hundred are known to occur. Many Hilbert series correspond to more than one family of QQ-Fano 3-folds having different Betti numbers. The talk will discuss some of the basic ideas in this classification, and outline some methods that can give useful information on different corners of this diverse kingdom.