A stabilized semidiscrete finite element discretization of the transient transport equation is studied in the framework of anisotropic meshes. A priori and a posteriori error estimates are derived, the involved constants being independent of the mesh aspect ratio, only space discretization being considered. Numerical results on nonadapted, anisotropic meshes and small time steps confirm the sharpness of the theoretical predictions. An anisotropic, adaptive finite element algorithm is then proposed with the goal to control the L-2 error in space at final time, the time step being kept constant. Numerical results are then presented on anisotropic, adapted meshes and small time steps. Three different methods are proposed to interpolate the solution between two adapted meshes. Numerical results indicate that both the conservative interpolation method of [F. Alauzet and M. Mehrenberger, Internat. J. Numer. Methods Engrg., 84 (2010), pp. 1552-1588] or the L-2 projection advocated in [P. E. Farrell et al., Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 2632-2642] yield accurate results.