A charged particle beam travelling at the speed of light produces large electromagnetic wake fields which, through interactions with its surroundings, act back on the particles in the beam. This coupled system may become unstable, resulting in a deterioration of the beam quality. Such effects play a major role in most existing storage rings, as they limit the maximum performance achievable. In a collider, the presence of a second beam significantly changes the dynamics, as the electromagnetic interactions of the two beams on each other are usually very strong and may, also, limit the collider performances. This thesis treats the coherent stability of the two beams in a circular collider, including the effects of the electromagnetic wake fields and of the beam-beam interactions, with particular emphasis on CERN's Large Hadron Collider. As opposed to other colliders, this machine features a large number of bunches per beam each experiencing multiple long-range and head-on beam-beam interactions. Existing models describing the beams stability need to be extended accordingly. The beam-beam interactions are very non-linear, as a result, different particles in the beams oscillates with slightly different frequencies, which generates Landau damping of coherent motion. A numerical integrator is developed in order to estimate the effect of Landau damping on potentially unstable coherent modes of oscillation. As the characteristics of the machine and the beams vary along the operational cycle of the Large Hadron Collider, the beam-beam forces and consequently the effect of Landau damping may vary significantly. It is shown that some configurations are particularly critical. Measurements of coherent instabilities during these operational phases, as well as in dedicated experiments, are presented and compared to the model. While most observations are in agreement with the model, some observed instabilities remain unexplained. A mechanism is proposed as an explanation for these instabilities. It is shown that the distribution of particles in the beam can be distorted in the presence of external noise and that non-measurable variations of the distribution functions can lead to significant deterioration of the Landau damping strength. Multiparticle tracking simulations are used to demonstrate this effect in realistic conditions. The coherent modes of oscillation of the coupled system including the two beams and the electromagnetic wake fields is discussed by extending an existing linear model introduced earlier to describe single bunches encountering a single beam-beam interaction in order to describe the beams in the Large Hadron Collider, i.e. involving multiple bunches and multiple beam-beam interactions. It is shown that under certain circumstances, a resonant action of the beam-beam forces and the electromagnetic wake fields may lead to strong instabilities. An experiment was performed to test these effects in the Large Hadron Collider, showing a good agreement with the models. Mitigation techniques against these instabilities are investigated, the effect of Landau damping is discussed by comparing the linear model with multiparticle tracking simulations. An operational solution to relax the limitations due to coherent instabilities, taking advantage of the stabilising effect of the head-on beam-beam interactions, is discussed based on the models developed and tested experimentally.