In this talk, following [Far1] and [Far2], we will explore semistability and Harder-Narasimhan filtrations of p-divisible groups over a complete rank 1 valuation ring of mixed characteristic. We begin by discussing the Harder-Narasimhan theory for finite flat group schemes and reviewing its main properties. We then turn to p-divisible groups. Although there is no "honest" Harder-Narasimhan theory in this setting as there is for finite flat group schemes, one can still meaningfully define concepts such as semistability and Harder-Narasimhan polygons. We will then look at the class of p-divisible groups of HN-type, which are those admitting a Harder-Narasimhan filtration, and discuss how general p-divisible groups can be "approximated" by such groups in a suitable sense. Finally, we will see that when the base ring is a DVR, any p-divisible group is isogenous to one of HN-type. This result will be a key input for the second talk, in which we will show how this semistability theory can be used to study the geometry of Rapoport-Zink spaces.