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We review several results related to the problem of a quantum particle in arandom environment.  In an introductory part, we recall how several functionals of the Brownianmotion arise in the study of electronic transport in weakly disordered metals(weak localization).  Two aspects of the physics of the one-dimensional strong localization arereviewed : some properties of the scattering by a random potential (time delaydistribution) and a study of the spectrum of a random potential on a boundeddomain (the extreme value statistics of the eigenvalues).  Then we mention several results concerning the diffusion on graphs, and moregenerally the spectral properties of the Schr\"odinger operator on graphs. Theinterest of spectral determinants as generating functions characterizing thediffusion on graphs is illustrated.  Finally, we consider a two-dimensional model of a charged particle coupled tothe random magnetic field due to magnetic vortices. We recall the connectionbetween spectral properties of this model and winding functionals of the planarBrownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and  conclusion added. Several references adde

Functionals of Brownian motion, localization and metric graphs