We survey some of the recent developments in the study of logarithm laws and
shrinking target properties for various families of dynamical systems. We
discuss connections to geometry, diophantine approximation, and probability
theory.Comment: This is a survey paper written following the Conference on Measures
  and Dyanmics on groups and homogeneous spaces, at TIFR, Mumbai, in Dec. 2007.
  It is in honor of Prof. S.G. Dani's 60th Birthda

Athreya, Jayadev S.