Given a normally hyperbolic invariant manifold $\Lambda$ for a map$f$, whose stable and unstable invariant manifolds intersecttransversally, we consider its associated scattering map. Thatis, the map that, given an asymptotic orbit in the past, gives theasymptotic orbit in the future.We show that when $f$ and $\Lambda$ are symplectic (resp. exactsymplectic) then, the scattering map is symplectic (resp. exactsymplectic). Furthermore, we show that, in the exact symplecticcase, there are extremely easy formulas for the primitivefunction, which have a variational interpretation as difference ofactions.We use this geometric information to obtain efficient perturbativecalculations of the scattering map using deformation theory. Thisperturbation theory generalizes and extends several resultsalready obtained using the Melnikov method. Analogous resultsare true for Hamiltonian flows. The proofs are obtained bygeometrically natural methods and do not involve the use ofparticular coordinate systems, hence the results can be used toobtain intersection properties of objects of any type.We also reexamine the calculation of the scattering map in ageodesic flow perturbed by a quasi-periodic potential. We showthat the geometric theory reproduces the results obtained in[Delshams et al., Adv. Math., 202(1):64-188,2006] using methods offast-slow systems. Moreover, the geometric theory allows tocompute perturbatively the dependence on the slow variables, whichdoes not seem to be accessible to the previous methods.