  Explicit birational geometry of 3-folds represents a second phase of Mori
theory, going beyond the foundational work of the 1980s. This paper is a
tutorial and colloquial introduction to the explicit classification of Fano
3-folds (Q-Fano 3-folds), a subject that we hope is nearing completion. With
the intention of remaining accessible to beginners in algebraic geometry, we
include examples of elementary calculations of graded rings over curves and K3
surfaces. For us, K3 surfaces have at worst Du Val singularities and are
polarised by an ample Weil divisor; they occur as the general elephant of a
Fano 3-fold. A second section of the paper runs briefly through the classical
theory of nonsingular Fano 3-folds and Mukai's extension to indecomposable
Gorenstein Fano 3-folds. Ideas sketched out by Takagi at the Singapore
conference reduce the study of Q-Fano 3-folds with g>=2 to indecomposable
Gorenstein Fano 3-folds together with unprojection data.
  Much of the information about the anticanonical ring of a Fano 3-fold or K3
surface is contained in its Hilbert series. The Hilbert function is given by
orbifold Riemann--Roch (see Reid's Young Person's Guide); using this, we can
treat the Hilbert series as a simple collation of the genus and a basket of
cyclic quotient singularities. Many hundreds of families of K3s and Fano
3-folds are known, among them a large number with g<=0, and Takagi's methods do
not apply to these. However, in many cases, the Hilbert series already gives
firm indications of how to construct the variety by biregular or birational
methods. A final section of the paper introduces the K3 database in Magma, that
manipulates these huge lists without effort.
