The title refers to the area of research which studies infinite groups
using measure-theoretic tools, and studies the restrictions that group structure
imposes on ergodic theory of their actions.
The paper is a survey of recent developments focused on the notion of Measure Equivalence
between groups, and Orbit Equivalence between group actions.
We discuss known invariants and classification results (rigidity)
in both areas.
The paper is dedicated to Robert Zimmer on the occasion of his 60th birthday.