In the recent biomechanical theory of cancer growth,solid tumors are considered as liquid-like materials comprising elastic components.In this fluid mechanical view,the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate,with the latter depending on the local cell density(contact inhibition) or/and on the mechanical stress in the tumor.For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling,the authors prove that there are always traveling waves above a minimal speed,and analyse their shapes.They appear to be complex with composite shapes and discontinuities.Several small parameters allow for analytical solutions,and in particular,the incompressible cells limit is very singular and related to the Hele-Shaw equation.These singular traveling waves are recovered numerically. In the recent biomechanical theory of cancer growth, solid tumors are considered as liquid-like materials comprising elastic components. This fluid mechanical view. The expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate, with the latter depending on the local cell density (contact inhibition) or / and on the mechanical stress in the tumor. For the two by two degenerate parabolic / elliptic reaction-diffusion system that results from this modeling, the authors prove that there are always traveling waves above a minimal speed, and analyze their shapes.They appear to be complex with composite shapes and discontinuities. Smallveral parameters allow for analytical solutions, and in particular, the incompressible cells limit is very singular and related to the Hele-Shaw equation. The sing singular traveling waves are recovered numerically.