Solid-liquid separation is an operation that starts with a dispersion of solid particles in a liquid and removes some of the liquid from the particles, producing a concentrated solid paste and a clean liquid phase. It is similar to thermodynamic processes where pressure is applied to a system in order to reduce its volume. In dispersions, the resistance to this osmotic compression depends on interactions between the dispersed particles.The first part of this work deals with dispersions of repelling particles, which are either silica nanoparticles or synthetic clay platelets, dispersed in aqueous solutions. In these conditions, each particle is surrounded by an ionic layer, which repels other ionic layers. This results in a structure with strong short-range order. At high particle volume fractions, the overlap of ionic layers generates large osmotic pressures; these pressures may be calculated, through the cell model, as the cost of reducing the volume of each cell. The variation of osmotic pressure with volume fraction is the equation of state of the dispersion. The second part of this work deals with dispersions of aggregated particles, which are silica nanoparticles, dispersed in water and flocculated by multivalent cations. This produces large bushy aggregates, with fractal structures that are maintained through interparticle surface-surface bonds. As the paste is submitted to osmotic pressures, small relative displacements of the aggregated particles lead to structural collapse. The final structure is made of a dense skeleton immersed in a nearly homogeneous matrix of aggregated particles. The variation of osmotic resistance with volume fraction is the compression law of the paste; it may be calculated through a numerical model that takes into account the noncentral interparticle forces. According to this model, the response of aggregated pastes to applied stress may be controlled through the manipulation of interparticle adhesion.