  The perceived randomness in the time evolution of "chaotic" dynamical systems
can be characterized by universal probabilistic limit laws, which do not depend
on the fine features of the individual system. One important example is the
Poisson law for the times at which a particle with random initial data hits a
small set. This was proved in various settings for dynamical systems with
strong mixing properties. The key result of the present study is that, despite
the absence of mixing, the hitting times of integrable flows also satisfy
universal limit laws which are, however, not Poisson. We describe the limit
distributions for "generic" integrable flows and a natural class of target
sets, and illustrate our findings with two examples: the dynamics in central
force fields and ellipse billiards. The convergence of the hitting time process
follows from a new equidistribution theorem in the space of lattices, which is
of independent interest. Its proof exploits Ratner's measure classification
theorem for unipotent flows, and extends earlier work of Elkies and McMullen.
