Nonadiabatic Quantum Chemistry is a nice Chemical Reviews article by David Yarkony.
It is quite succinct but covers a significant number of specific chemical systems where non-adiabatic effects [including conical intersections] are important and have been treated theoretically.
Here I just mention one example for which theory has failed so far, the vibrationally mediated photodissociation of NH3 (ammonia) to NH2 + H.
Experiments find that if the excited state contains a symmetric (asymmetric) N-H stretch the dominant decay channel is to the NH2 ground state (excited state). Yarkony says that calculations [e.g. this one from Truhlar's group, which contains the figure below] have not yet captured this vibrational selectivity.