Hydrodynamic interactions in a suspension of spherical particles confined between two parallel planar walls are studied under creeping-flow conditions. The many-particle friction matrix in this system is evaluated using our novel numerical algorithm based on transformations between Cartesian and spherical representations of Stokes flow. The cartesian representation is used to describe the interaction of the fluid with the walls and the spherical representation is used to describe the interaction with the particles. The transformations between these two representations are given in a closed form, which allows us to evaluate the coefficients in linear equations for the induced-force multipoles on particle surfaces. the friction matrix is obtained from these equations, supplemented with the superposition lubrication corrections. we have used our algorithm to evaluate the friction matrix for a single sphere, a pair of spheres, and for linear chains of spheres. The friction matrix exhibits a crossover from a quasi-two-dimensional behaviour (for systems with small wall separation H) to THE three-dimensional behaviour (when the distance H is much larger than the interparticle distance L). the crossover is especially pronounced for a long chain moving in the direction normal to its orientation and parallel to the walls. in this configuration, a large pressure build-up occurs in front of the chain for small values of the gapwidth H, which results in a large hydrodynamic friction force. a standard wall superposition approximation does not capture this behaviour.