Active particle suspensions, of which a bath of swimming bacteria is a paradigmatic example, are characterized by complex dynamics involving strong fluctuations and large-scale correlated motions. These motions, which result from the many-body interactions between particles, are biologically relevant as they impact mean particle transport, mixing and diffusion, with possible consequences for nutrient uptake and the spreading of bacterial infections. To analyze these effects, a kinetic theory is presented and applied to elucidate the dynamics and pattern formation arising from mean-field interactions. Based on this model, the stability of both aligned and isotropic suspensions is investigated. In aligned suspensions, an instability is shown to always occur at finite wavelengths, in agreement with previous predictions and simulations. In isotropic suspensions, a new instability for the active particle stress is also found to exist, in which shear stresses are eigenmodes and grow exponentially at low wavenumbers, resulting in large-scale fluctuations in suspensions of pusher particles above a threshold concentration. Numerical simulations of the kinetic equations are also performed, and applied to study the long-time nonlinear dynamics, which are characterized by transient particles clusters that form and break up in time, as well as complex chaotic flows correlated on the system size. The predictions from the kinetic model are then tested using direct particle simulations accounting for multi-body hydrodynamic interactions between model microswimmers: these simulations confirm the existence of a transition to correlated motions and large-scale flows above a certain volume fraction, as demonstrated by a sharp increase in density fluctuations, velocity correlation lengths, and mean particle velocities. The effect of this transition on fluid mixing is also investigated, and the emergence of large-scale flows is shown to significantly enhance convective mixing. To conclude, consequences of particle activity on the effective rheology of the suspensions are briefly discussed. We demonstrate that the rheology is characterized by much stronger normal stress differences than in passive suspensions, and that tail-actuated swimmers result in a strong decrease in the effective shear viscosity of the fluid.