Analyzing the phase transition from diffusive to localized behavior in a model of directed
polymers in a random environment this volume places particular emphasis on the localization
phenomenon. The main questionis: What does the path of a random walk look like if rewards and
penalties are spatially randomly distributed?This model which provides a simplified version of
stretched elastic chains pinned by random impurities has attracted much research activity but
it (and its relatives) still holds many secrets especially in high dimensions. It has
non-gaussian scaling limits and it belongs to the so-called KPZ universality class when the
space is one-dimensional. Adopting a Gibbsian approach using general and powerful tools from
probability theory the discrete model is studied in full generality. Presenting the
state-of-the art from different perspectives and written in the form of a first course on the
subject this monograph is aimed at researchers in probability or statistical physics but is
also accessible to masters and Ph.D. students.
