The fractional quantum Hall effect is among the most exquisite, most consequential, and best understood manifestations of strong correlations in nature. I will begin with a brief introduction to its rich phenomenology and its explanation as a consequence of the formation of exotic particles called composite fermions, which are born when electrons effectively swallow a part (or whole) of the magnetic field. I will then report on three types of bound states of composite fermions and their experimental manifestations: Cooper pairs of composite fermions (responsible for fractional quantum Hall states that are thought to harbor “non-Abelian anyons”), the complex multi-composite fermion bound states that resemble an electron (probed by scanning tunneling microscopy), and the chiral spin-2 composite-fermion particle-hole pair (observed in Raman experiments with circularly polarized photon). The talk will be presented at a level that should be accessible to graduate students and non-experts.