The study of resonances and chaos within space debris dynamics is of paramount importance for the analysis of the global behavior of small objects orbiting around the Earth. In particular, it is important to understand the location of the equilibria, the existence of regular or chaotic regions, and especially their behavior as some control parameters are varied. These studies might allow to develop mitigation, maintenance and control strategies based on mathematical investigations. Numerical and analytical techniques can be used to investigate special resonances affecting the dynamics of space debris in different regions of the space around our planet. This allows to study the stability of the orbital elements and the occurrence of regular and chaotic motions.