The classical dynamics of an electron moving in the presence of two equally charged, fixed nuclei is presented. The manner in which the values of the three constants of the motion determine the qualitative features of the electronic trajectory is discussed. Primitive semiclassical wavefunctions and quantization conditions for the Born-Oppenheimer electronic quantum states of H2+ are derived using the canonically invariant quantization methods of Einstein, Keller, and Maslov. Because of the presence of energetic and dynamical barriers, a uniform quantization method is needed to give quantitative results for all internuclear separations. We employ a well-established uniformization which models the effective potential barrier as a parabola. Finally, the eigenparameters computed using the primitive and uniform quantization methods are compared with exact Born-Oppenheimer quantum mechanical results for the six lowest Σ states, the two lowest Π states, and the two lowest ∆ states for a wide range of internuclear separations. The electronic energies computed using the uniform quantization conditions typically agree with the exact quantum results to within a fraction of a percent.