The periodic points of a discrete dynamical system control its local and global dynamical behaviour. When the system is defined over the rational numbers, one can ask about the arithmetic properties of periodic points. The central open conjecture in arithmetic dynamics asks whether there are uniform constraints on the possible periods of points for families of algebraic dynamical systems. In this talk, we will discuss this conjecture, how it generalizes the torsion conjecture—in particular, the celebrated theorems of Mazur and Merel on rational torsion of elliptic curves—and survey some recent progress on and strategies for attacking this problem.