A review of literature on the flow of biofluids in compliant tubes, ducts and cavities is presented. Some common features connected with fluid-structure interaction and system instability are described. The corresponding mathematical models, theoretical and numerical results are given. Included is some pioneering research that has been published in Russian but never been translated into English. Original results on blood flow in vessels modeled as multilayered, anisotropic, viscoelastic tubes are presented. Depending on the boundary condition imposed at the outer surface of a tube, it is shown that such systems can be stabilized by judicious choices of the viscous and elastic parameters of its separate layers. The obtained results can be utilized to achieve flow stabilization in compliant ducts and tubes of technical and biomedical devices.