Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of positive characteristic $p$. We will first define several subtle invariants of $D$ which have been introduced recently and which are crucial for any strong, refined classification of $D$. Then we will present our results on them. For instance, two deep conjectures of Traverso from mid 1970's pertain to the smallest truncations $D[p^m]$ of $D$ which determine either the Newton polygon of $D$ or even the isomorphism class of $D$. We report on two joint works. The first is with Ofer Gabber and the second with Marc-Hubert Nicole and Eike Lau. Based on results of the first work, the second proves refined versions of these two conjectures. If time permits, applications of our results and their proofs will be as well mentioned. The talk will be accessible to graduate students that have some familiarity with the ring of Witt vectors with coefficients in $k$.