The high luminosity foreseen in the future electron-positron circular collider (FCC-ee) necessitates very intense multi-bunch colliding beams with very small transverse beam sizes at the collision points. This requires transverse beam emittances comparable to those of modern synchrotron light sources. At the same time, the stored beam currents should be close to the best values achieved in the last generation of particle factories. This combination of demanding factors poses a major challenge, namely preserving a high beam quality, while, at the same time, avoiding machine performance degradation. In consequence, a careful study of the collective effects and identification of stabilizing mechanisms are required to mitigate the foreseen instabilities. In this contribution, we discuss the current status of these studies