A foam having inter-communicating fine pores throughout has a high liquid absorbing power and is rich in liquid-holding ability. When a slurry-state liquid is poured onto the surface of a liquid-permeable filter structure provided with such a porous layer having a good elastic recoverability from its compressed state, the liquid part of said slurry is absorbed in the foam structure due to the capillary action of the pores, whereas the mud-like solid part (wet cake) suspended in the slurry is retained on the surface of the porous layer in coagulated fashion and thus it is separated from the liquid part. By positioning a transfer means having a substantially even and smooth work surface so as to be in appropriate contact with the resulting layer of the solid part deposited on the sheet surface, the said solid part is transferred from the said sheet surface onto the work surface of the transfer means and is deposited thereon. Therefore, by causing this sheet structure to make continuous circulatory movement and also causing the transfer means which is positioned close thereto to move continuously at roughly the same speed and in the direction corresponding to the moving filter structure, and by continuously pouring the slurry-state liquid onto the foam surface, it is possible to continuously separate the said slurry into the solid part and the liquid part with a high efficiency. If the separating apparatus is arranged so that the filter structure is mounted on the circumference of a rotary drum body, the separating apparatus will conveniently have a compact size, requiring only a small floor area and space for its installation.