  We provide a self-contained, accessible introduction to Ratner's
Equidistribution Theorem in the special case of horocyclic flow on a complete
hyperbolic surface of finite area. This equidistribution result was first
obtained in the early 1980s by Dani and Smillie and later reappeared as an
illustrative special case of Ratner's work on the equidistribution of unipotent
flows in homogeneous spaces. We also prove an interesting probabilistic result
due to Breuillard: on the modular surface an arbitrary uncentered random walk
on the horocycle through almost any point will fail to equidistribute, even
though the horocycles are themselves equidistributed. In many aspects of this
exposition we are indebted to Bekka and Mayer's more ambitious survey, "Ergodic
Theory and Topological Dynamics for Group Actions on Homogeneous Spaces."
