In studies on geophysical fluid dynamics, it is common practice to take the Coriolis force only partially into account by neglecting the components proportional to the cosine of latitude, the so-called traditional approximation (TA). This review deals with the consequences of abandoning the TA, based on evidence from numerical and theoretical studies and laboratory and field experiments. The phenomena most affected by the TA include mesoscale flows (Ekman spirals, deep convection, and equatorial jets) and internal waves. Abandoning the TA produces a tilt in convective plumes, produces a dependence on wind direction in Ekman spirals, and gives rise to a plethora of changes in internal wave behavior in weakly stratified layers, such as the existence of trapped short low-frequency waves, and a poleward extension of their habitat. In the astrophysical context of stars and gas giant planets, the TA affects the rate of tidal dissipation and also the patterns of thermal convection.