As accelerator technology advances, the requirements on accelerator beam quality become increasingly demanding. Facing these new demands, the topic of phase space gymnastics is becoming a newly-focused and critical area of R&D. In a phase space gymnastic, the beam’s phase space distribution is manipulated and precision tailored to meet the required beam quality. On the other hand, all realization of such gymnastics will have to obey accelerator physics principles as well as technological limitations. Recent examples of phase space gymnastics include Emittance exchanges, Phase space exchanges, Emittance partitioning, Seeded FELs and Microbunched radiators. In this talk, we will review the emittance related topics. The accelerator physics basis, the lattice designs that provide these phase space manipulations, and the possible applications of these gymnastics, will be discussed. This fascinating field promises to be a focused attention towards the future.