  The model of directed polymer in a random environment is a fundamental model
of interaction between a simple random walk and ambient disorder. This
interaction gives rise to complex phenomena and transitions from a central
limit theory to novel statistical behaviours. Despite its intense study, there
are still many aspects and phases which have not yet been identified. In this
review we focus on the current status of our understanding of the transition
between weak and strong disorder phases, give an account of some of the methods
that the study of the model has motivated and highlight some open questions.
