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When averaged on a few thousands years, the Earth΄s magnetic field can be roughly described as the one of an axial dipole. As shown by paleomagnetic records, it has frequently reversed its polarity on geological time scales. Field reversals have also been reported in several numerical simulations of the geodynamo and more recently, in a laboratory experiment involving a von Karman swirling flow of liquid sodium (VKS). We first recall some of the main experimental results and understand them using phenomenological models and numerical simulations. In particular, we show that all the regimes of the magnetic field observed in the experiments reported so far, can be understood in the framework of low dimensional dynamical system theory: two modes of the magnetic field, with dipolar (respectively quadrupolar) symmetry, can be generated by the turbulent flow of liquid sodium, and their interaction can lead to the observed dynamics (oscillations, random reversals, symmetric or asymmetric bursts). Turbulent fluctuations alone do not drive these dynamical regimes that only occur when a symmetry of the flow is broken. Although the flow in the Earth΄s core strongly differs from the one of the VKS experiment, a similar model but based on a different broken symmetry, can be used. It explains several features observed in paleomagnetic recordings of the Earth΄s magnetic field reversals. [ABSTRACT FROM AUTHOR]