This paper presents an adaptive finite element method for solving scalar hyperbolic transport problems. An equation for the evolution of the error is developed. The Lesaint-Raviart finite element method is used to solve both the transport problem and the error equation. The use of a hierarchical finite element basis on triangles leads to a very efficient error estimation algorithm. The adaptive strategy, based on remeshing, is dmonstrated on several non-trivial problems with known analytical solutions. Higher degree polynomials combined with adaptation produce a very efficient solution algorithm even for problems involving discontinuities.