The perturbative description of particle accelerators using Taylor series maps is discussed. A new technique is presented which allows a very efficient computation of high order maps. It is shown how this method can be used to determine maps of the action of various electromagnetic fields including fringe fields, measured fields, and space-charge fields.Once the map of the system is known, it can be used for several purposes. Firstly, it allows an exact, non-numerical calculation of quantities of interest like tune shifts and chromaticities. Secondly, it can be tranformed into different coordinates in which the motion of the particles has a particularly simple form and which is helpful for the search of invariants and KAM surfaces. Finally the map can also be used for tracking, if desired also in combination with symplectification procedures. Here the major advantage of the map is the gain in speed compared to direct tracking. As with any tracking method, however, care should be exercised not to overestimate the results of such simulations, in particular for large numbers of iterations.