We study the propagation of rays, paroxial rays, and Gaussian beams in a medium where slowness differs only slightly from that of a reference medium. Ray theory is developed using a Hamiltonian formalism that is independent of the coordinate system under consideration. Let us consider a ray in the unperturbed medium. The perturbation is slowness produces a change of the trajectory of this ray which may be calculated by means of canonical perturbation theory. We define paraxial rays as those rays propagate in perturbed medium in the vicinity of the perturbed ray. The ray tracing equation for paroxial rays may be obtained by a linearisation of the canoncial ray equations. The linearized equations are then solved by a propagator method. With the help of the propagator we form beams, i.e. families of paraxial rays that depend on a single beam parameter. The results are very general and may be applied to a number of kinematic and dynamic ray tracing problems, like two-point ray tracing, Gaussian beams, wave front interpolation, etc. The perturbation methods are applied to the study of a few simple problems in which the unperturbed medium is homogeneous. First, we consider a two-dimensional spherical inclusion with a Gaussian slowness perturbation profile. Second, transmission and reflection problems are examined. We compare results for amplitude and travel time computed by exact and perturbed ray theory. The agreements is excellent and may be improved using an iterative procedure by which we change the reference unperturbed ray whenever the perturbation becomes large. Finally, we apply our technique to a three-dimensional problem: we calculate the amplitude perturbation and ray deflection produced by the velocity structure under the Mont Dore volcano (central France). Again a comparison shows excellent agreement between exact and perturbed ray theory. ¿ American Geophysical Union 1987