In the last decade there have been several important breakthroughs in Number Theory and Diophantine Geometry, where progress on long-standing open problems has been achieved by utilising ideas originated in the theory of dynamical systems on homogeneous spaces. Dynamical systems techniques are applicable to a wide range of number-theoretic objects that have many symmetries. In particular: various question in Diophantine approximation have been studied using recurrence properties of flows on the space of unimodular lattices diagonal flows have played an important role in recent advances on...