This chapter is build from three 1.5 hours lectures given in Udine in april 2018 on various aspects of Earth's core dynamics. The chapter starts with a short historical note on the discovery of Earth's magnetic field and core (section 1). We then turn to an introduction of magnetohydrodynamics (section 2), introducing and discussing the induction equation and the form and effects of the Lorentz force. Section 3 is devoted to the description of Earth's magnetic field, introducing its spherical harmonics description and showing how it can be used to demonstrate the internal origin of the geomagnetic origin. We then move to an introduction of the convection-driven model of the geodynamo (section 4), discussing our current understanding of the dynamics of Earth's core, obtaining heuristically the Ekman dependency of the critical Rayleigh number for natural rotating convection, and introducing the equations and non-dimensional parameters used to model a convectively driven dynamo. The following section deals with the energetics of the geodynamo (section 5). The final two section deal with the dynamics of the inner core, focusing on the effect of the magnetic field (section 6), and with the formation of the core (section 7). Given the wide scope of this chapter and the limited time available, this introduction to Earth's core dynamics is by no means intended to be comprehensive. For more informations, the interested reader may refer to Jones (2011), Olson (2013), or Christensen and Wicht (2015) on the geomagnetic field and the geodynamo, to Sumita and Bergman (2015), Deguen (2012) and Lasbleis and Deguen (2015) on the dynamics of the inner core, and to Rubie et al. (2015) on core formation.