This thesis aims at understanding the internal dynamics of solar-type stars and the origin of their magnetism. We will explore the complex nonlinear interactions between convection, rotation and magnetism conducting both 2D (STELEM code) and 3D (ASH code) numerical simulations. This dual approach will unveil the mechanisms and key parameters behind those physical processes. While the Sun has played a central role in previous studies, this work extends our knowledge to G and K stars. This manuscript is divided into 4 parts. The first one introduces the concepts behind internal stellar dynamics, and emphasizes the dynamo effect. Accurate observations of the Sun will be compared to stellar data, allowing us to determine what is specific to the Sun and what is generic for all stars. The second part reports the results obtained with the 2D STELEM code. This code allows us to study the generation and evolution of the large scale magnetic fields on a timescale comparable to the solar cycle period (11 years), giving us insight into the underlying dynamo processes at work. We show that the current solar models cannot reproduce the observations, when applied to rapidly rotating stars, unless we consider a turbulent pumping mechanism under specific conditions. Then, we have improved these kinematic models by taking into account the large scale magnetic field feedback on the longitudinal velocity component, called the Malkus Proctor effect. The models are now able to reproduce the solar torsional oscillations and can predict how their properties evolve with rotation rate. The third part focuses on 3D numerical simulations running on massively parallel supercomputers, using the ASH code. In constrast with the previously described code, ASH explicitly resolves the full MHD equations. We have studied (hydrodynamically) how the convective properties of G and K stars change as function of mass and rotation rate, first by considering the convective envelope alone, then by taking into account the dynamical coupling with the underlying radiative zone. We show that the dynamics is mainly driven by the Rossby number, and that the flow caracteristics qualitatively change when the Rossby number is above unity. We also derive scaling laws for the global flows properties (differential rotation, meridional circulation, etc.) as function of mass and rotation rate. The last part sums up the main results and the perspectives of the work presented above. We are developing 3D numerical simulations involving magnetic dynamo action in rapidly rotating stars. In these stars, the magnetic field is self-organized in strong wreathes close to the equator extracting its energy from the convective kinetic energy but also from the strong differential rotation. Finally, we explain the benefits of a combined approach that uses both 2D and 3D models.
