Philosophical 3-folds: how Tom and Jerry became existentialists

The history of philosophy is full of deep and unanswered questions. Who are we? What is our purpose in this world? How many QQ-Fano threefolds exist in high codimension? Prophets Mori and Mukai in the late 80s provided a classification of smooth Fano threefolds of index one, and their 17+88 families are an irreplaceable tool in every birational geometer working kit. More important than the list itself was the strategy (the 'vector bundle method') and the mantra: Fano threefolds arise as linear section of "good" varieties coming from representation theory. The same results sadly does not extends to the QQ-Fano threefolds case (the natural playground for MMPers). Tom and Jerry models provides a good level of understanding up to codimension three, but the garden is still wild and untamed. In this talk we will provide an overview of the subject, and we will report on some recent progress in the classification.