Gradient fields are irrotational. Pressure-robust mixed finite element methods for the incompressible Stokes equations fulfill this property exactly. The talk will illustrate and explain the significance of pressure-robustness in incompressible and compressible flow problems, by comparing pressure-robust mixed methods with classical mixed methods. Physical regimes are identified, when pressure-robustness matters. Nonlinearities are shown to trigger the need for pressure-robustness.