We consider classical versions of three generalizations of transition state theory: microcanonical variational transition state theory, canonical variational transition state theory, and Miller's unified statistical theory. We prove that microcanonical variational transition state theory is identical with the adiabatic theory of reactions, i.e., to adiabatic transition state theory. To test the validity of these approximate theories, we present calculations for several collinear reactions of hydrogen and halogen atoms with hydrogen molecules. Average reaction probabilities are computed using conventional and microcanonical variational transition state theory and the unified statistical theory and are compared with those of exact classical dynamics for seven cases. These results confirm the general validity of the fundamental assumption of transition state theory at low energy and show that the variational method can be used to extend the range of validity to higher energies. Thermal rate constants are calculated by these methods and by canonical variational transition state theory for nine systems. Using a Morse approximation involving the second and third derivatives of the local vibrational well at its minimum, the average absolute value of the error and range of absolute values of the errors at 600 K for the seven cases where we computed exact classical canonical rate constants are 28 and 0-78% for conventional transition state theory, 10 and 1-37% for the microcanonical variational theory or adiabatic transition state theory, 7 and 1-22% for the unified statistical theory, and 15 and 0-41% for the canonical variational theory.