The basic theoretical tools for the calculation of cross sections in ion-atom and ion-molecule collisions are almost all based on the close-coupling (or coupled-state) method. For slow collisions where the projectile velocity is small compared to the typical velocity of the outer electrons, the so-called Perturbed Stationary State (PSS) approximation originally proposed more than half a century ago is usually the main starting point. In the PSS model, the total wavefunction of the system is expanded in terms of molecular orbitals of the total collision system, similar to the Born-Oppenheimer approximation for molecules. Despite of its wide usuage, the PSS model is known to suffer from many serious defects, all originating from the fact that the molecular orbitals used in the PSS model do not satisfy correct asymptotic boundary conditions. For collisions where the motion of the heavy particles is treated classically, in the past two to three decades, a number of remedies have been proposed -- notably in the form of attaching electron translational factors to each molecular orbital, thus maintaining the structure of the close-coupling equations. For much lower energies where the motion of the heavy particles is treated quantum mechanically, the so-called reaction coordinates have been used in theoretical calculations. The use of electron translational factors and reaction coordinates, despite of the fact that it removes the main defects of the PSS approximation, does suffer from the criticism that they are not derived from first principles and their introduction is somewhat arbitrary. In this talk I will discuss a new formulation of the simplest ion-atom collisions based on the hyperspherical close-coupling method [1]. In its formulation, it is similar to the PSS approximation, but without its inherent defects. The results of the hyperspherical close-coupling calculations will be compared to those from using the reaction coordinates to confirm that the results from the latter are indeed reliable. Thus we believe that data for ion-atom and ion-molecule collisions needed for applications can be obtained with confidence if proper care has been taken in the theoretical calculations. Due to the complexity of the calculation, in the literature the latter is by no means guaranteed. We will also discuss theoretical results based on close coupling calculations using atomic orbitals as basis functions, and those based on the hidden crossing theory.

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