The classical field-dependent parametrization covariant Hamiltonian formulation of the open and the closed string is discussed. The formalism is not applicable to the open string. A conformally covariant formalism is developed for the open string. The Rohrlich gauge conditions are justified and applied. The parametrization of classical solutions is not uniquely fixed; the generators of rigid time translation in the parameter space remain first class. The constraints and gauge conditions are taken into account in the quantum theory as conditions on physical states. The required invariance of physical states under rigid displacement of parameter time leads to a mass superselection rule. The set of physical string quantum states is analogous to the set of states constructed by Di Vecchia, Del Guidice, and Fubini. A recursive construction is presented which permits the counting of physical states of any given mass, spin, and parity. Physical states lie on linearly rising Regge trajectories with one universal slope. The intercept of the leading trajectory is constrained only by the requirement that there be no tachyonic physical states. The quantization is carried out in four space-time dimensions.