Our purpose in this paper is to provide the framework for a generalization of classical mechanics and electrodynamics, including Maxwell's theory, which is simple, technically correct, and requires no additional work for the quantum case. We first show that there are two other definitions of proper-time, each having equal status with the Minkowski definition. We use the first definition, called the proper-velocity definition, to construct a transformation theory which fixes the proper-time of a given physical system for all observers. This leads to a new invariance group and a generalization of Maxwell's equations left covariant under the action of this group. The second definition, called the canonical variables definition, has the unique property that it is independent of the number of particles. This definition leads to a general theory of directly interacting relativistic particles. We obtain the Lorentz force for one particle (using its proper-time), and the Lorentz force for the total system (using the global proper-time). Use of the global proper-time to compute the force on one particle gives the Lorentz force plus a dissipative term corresponding to the reaction of this particle back on the cause of its acceleration (Newton's third law). The wave equation derived from Maxwell's equations has an additional term, first order in the proper-time. This term arises instantaneously with acceleration. This shows explicitly that the longsought origin of radiation reaction is inertial resistance to changes in particle motion. The field equations carry intrinsic information about the velocity and acceleration of the particles in the system. It follows that our theory is not invariant under time reversal, so that the existence of radiation introduces an arrow for the (proper-time of the) system.