  We study the growth of correlations in systems with weak long-range
interactions. Starting from the BBGKY hierarchy, we determine the evolution of
the two-body correlation function by using an expansion of the solutions of the
hierarchy in powers of 1/N in a proper thermodynamic limit $N\to +\infty$.
These correlations are responsible for the ``collisional'' evolution of the
system beyond the Vlasov regime due to finite $N$ effects. We obtain a general
kinetic equation that can be applied to spatially inhomogeneous systems and
that takes into account memory effects. These peculiarities are specific to
systems with unshielded long-range interactions. For spatially homogeneous
systems with short memory time like plasmas, we recover the classical Landau
(or Lenard-Balescu) equations. An interest of our approach is to develop a
formalism that remains in physical space (instead of Fourier space) and that
can deal with spatially inhomogeneous systems. This enlightens the basic
physics and provides novel kinetic equations with a clear physical
interpretation. However, unless we restrict ourselves to spatially homogeneous
systems, closed kinetic equations can be obtained only if we ignore some
collective effects between particles. General exact coupled equations taking
into account collective effects are also given. We use this kinetic theory to
discuss the processes of violent collisionless relaxation and slow collisional
relaxation in systems with weak long-range interactions. In particular, we
investigate the dependence of the relaxation time with the system size and
provide a coherent discussion of all the numerical results obtained for these
systems.
