  We classify the graphs that can occur as the graph of rational preperiodic
points of a quadratic polynomial over $\bold Q$, assuming the conjecture that
it is impossible to have rational points of period $4$ or higher. In
particular, we show under this assumption that the number of preperiodic points
is at most~$9$. Elliptic curves of small conductor and the genus~$2$ modular
curves $X_1(13)$, $X_1(16)$, and $X_1(18)$ all arise as curves classifying
quadratic polynomials with various combinations of preperiodic points. To
complete the classification, we compute the rational points on a non-modular
genus~$2$ curve by performing a $2$-descent on its Jacobian and afterwards
applying a variant of the method of Chabauty and Coleman.
