  In the past 20 years, compactifications of the families of curves in
algebraic varieties X have been studied via stable maps, Hilbert schemes,
stable pairs, unramified maps, and stable quotients. Each path leads to a
different enumeration of curves. A common thread is the use of a 2-term
deformation/obstruction theory to define a virtual fundamental class. The
richest geometry occurs when X is a nonsingular projective variety of dimension
3.
  We survey here the 13/2 principal ways to count curves with special attention
to the 3-fold case. The different theories are linked by a web of conjectural
relationships which we highlight. Our goal is to provide a guide for graduate
students looking for an elementary route into the subject.
