  This expository paper, based on a Current Events Bulletin talk at the
January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and
discusses the role they play in three areas: smooth ergodic theory,
Teichm\"uller theory, and the spectral theory of one-frequency Schr\"odinger
operators. The inspiration for this paper is the work of 2014 Fields Medalist
Artur Avila, and his work in these areas is given special attention.
