We consider several approximate methods in an attempt to correct two major deficiencies of the conventional transition state theory of chemical reactions. One deficiency is its overestimating thermal rate constants due to a breakdown of the fundamental transition state approximation that classical trajectories do not recross the transition stete dividing surface. This error classically becomes more important at high temperatures. The other shortcoming of conventional transition state theory is the neglect of quantal effects on the reaction-coordinate motion. This becomes more important at low temperature, and when tunneling is important conventional transition state theory underestimates the low-temperature rates, sometimes severely. We consider first three generalizations of transition state theory in which no quantal correction to the reaction-coordinate motion is included but bound motion normal to the reaction coordinate is quantized. These are canonical variational transition state theory, microcanonical variational transition state theory, which is equivalent to the (vibrationally) adiabatic theory of reactions, and Miller's unified statistical theory. We also consider several ways to make tunneling corrections. One is the Wigner transmission coefficient, and the other three involve one-mathematical-dimensional quantal scattering calculations performed numerically. The barriers for these calculations are the conservation of vibrational energy barrier, the minimum-energy-path vibrationally adiabatic barrier, and the Marcus-Coltrin-path vibrationally adiabatic barrier. To test the accuracy of these approximate methods, we present calculations for several collinear reactions of H, D, Cl, or I with five isotopes of hydrogen molecules and compare these results with those from accurate quantal calculations of the reaction probabilities as functions of energy and of the thermal rate constants as functions of temperature. Our results show that the variational theories and the unified statistical theory offer useful improvements over conventional transition state theory at high temperature although bounding inequalities are not satisfied as they are in purely classical transition state theory. No method of including one-dimensional quantal corrections for the reaction-coordinate motion was found to be adequate for all the systems studied. In fact for the I + H2 reaction on each of two potential energy surfaces the variational methods with classical treatment of the reaction-coordinate motion yield excellent results while all quantum corrections to the reaction-coordinate motion fail. For the other systems the vibrationally adiabatic method using the Marcus-Coltrin tunneling path gives the most consistently accurate results overall, although the Wigner correction gives more accurate results in some instances. A quantally unified statistical model with quantal corrections computed from the Marcus-Coltrin vibrationally adiabatic barrier predicts the most consistently accurate isotope effects for the H + H2 and Cl + H2 systems.