This work aims to study squeeze flows when the lubrication approximation does not necessarily hold. Strong squeeze flows are defined as the cases in which a sample is compressed by a disk with the initial speed of 40 cm/s, whereas weak squeeze flows are realized when the disk is softly released manually to avoid any impact of the sample at the beginning of compression. Strong and weak squeeze flows of yield-stress materials are studied experimentally and theoretically. In the experiments, disk-like constant-volume samples of Carbopol solutions and bentonite dispersions are compressed between two approaching disks being subjected to constant forces. In addition, experiments with shear flows in parallel-plate and vane viscometers are conducted. Using visualization through the transparent wall of the squeezing apparatus, it is demonstrated that the no-slip conditions hold. It is also demonstrated that during the fast stage of strong squeeze flows, the material response can be explained by deviatoric normal stresses, which elucidates the link of strong squeeze flows to elongational flows. The analysis of the data in the framework of the Newtonian and Herschel-Bulkley models shows that in the present case the nonlinearity of the rheological response at the fast stage of strong squeeze flows is not very significant, and a strain-rate-independent viscosity can be used as a reasonable approximation. On the other hand, at the final stage of squeeze flows, when samples spread significantly under the action of a constant squeezing force, the compressive stresses become small enough, and the dominant role is played by the yield stress. No significant signs of thixotrophy were observed. It is shown that strong squeeze flow in the squeezing apparatus is a convenient tool useful for the measurement of viscosity and the yield stress of complex soft materials. (C) 2013 The Society of Rheology. [http://dx.doi.org/10.1122/1.4794912]