Nearside-farside (NF) theory, as used to understand nuclear heavy-ion differential Gross sections, is applied for the first time to the angular scattering of atom-atom and atom-diatom collisions. A NF decomposition of the partial wave series (PWS) for the scattering amplitude has the following advantages : (a) it is exact, (b) it uses PW scattering matrix elements (quantum or semiclassical) as calculated by standard computer programs, (c) it is easily incorporated into existing computer programs which calculate angular distributions, (d) semiclassical techniques, such as stationary phase or saddle point integration, are not invoked for the PWS, although the semiclassical picture is still evident. A disadvantage of a NF decomposition is that it is not unique. The Fuller and Hatchell NF decompositions are used to analyze the angular scattering of four collision systems whose PWS involve Legendre polynomials : (a) atom-atom He+Ne elastic diffraction scattering, (b) atom-atom H++Ar elastic rainbow scattering, (c) atom rigid-rotator Ne+D-2(j=0) -->Ne+D-2(j) diffraction scattering under sudden conditions so that the infinite-order-sudden (IOS) approximation is valid, (d) atom rigid-rotator He+N-2(j=0)-->He+N-2(j) rotational rainbow IOS scattering. The utility of these two NF decompositions is assessed by comparison with results from the semiclassical complex angular momentum (CAM) representation of the scattering amplitude. This is chosen because it allows an unambiguous separation of the scattering amplitude into nearside and farside subamplitudes under semiclassical conditions. The Fuller NF decomposition, unlike the Hatchell NF decomposition, provides a physically clear explanation of the angular scattering, which always agrees with the semiclassical CAM interpretation (except for scattering angles approximate to 180 degrees). The Fuller NF decomposition is therefore recommended for applications to atomic and molecular collisions. The NF theory for the decomposition of Legendre polynomials is generalized to scattering amplitudes whose PWS involve associated Legendre functions or reduced rotation matrix elements.