If density of flow is globally a constant, then the flow is said incompressible. Otherwise, the flow is said compressible. Flow motion of compressible inviscid flow is governed by Euler system. The Euler system is a nonlinear PDE system describing conservation of mass, momentum and energy. Interestingly, mathematical property of Euler system dramatically changes depending on whether speed of flow exceeds sonic speed or not. In addition, nonlinearity of Euler system can cause a shock or a contact discontinuity/vortex sheet in which physical features of flow discontinuously change. This talk will present mathematical approach to understand such nonlinear physical phenomena. If time allows, some open problems will be discussed as well.