  This report discusses recent results as well as new perspectives in the
ergodic theory for Riemann surface laminations, with an emphasis on singular
holomorphic foliations by curves. The central notions of these developments are
leafwise Poincar\'e metric, directed positive harmonic currents, multiplicative
cocycles and Lyapunov exponents. We deal with various ergodic theorems for such
laminations: Random and Operator Ergodic Theorems, (Geometric) Birkhoff Ergodic
Theorems, Oseledec Multiplicative Ergodic Theorem and Unique Ergodicity
Theorems. Applications of these theorems are also given. In particular, we
define and study the canonical Lyapunov exponents for a large family of
singular holomorphic foliations on compact projective surfaces. Topological and
algebro-geometric interpretations of these characteristic numbers are also
treated.
  These results highlight the strong similarity as well as the fundamental
differences between the ergodic theory of maps and that of Riemann surface
laminations. Most of the results reported here are known. However, sufficient
conditions for abstract heat diffusions to coincide with the leafwise heat
diffusions (Subsection 5.2) are new ones.
