  These notes cover one of the topics programmed for the St Petersburg School
in Probability and Statistical Physics of June 2012.
  The aim is to review recent mathematical developments in the field of random
walks in random environment. Our main focus will be on directionally transient
and reversible random walks on different types of underlying graph structures,
such as $\mathbb{Z}$, trees and $\mathbb{Z}^d$ for $d\geq 2$.
