A handful of MSPs are detected in very compact orbits (orbital periods between 2 − 9 hr), around low-mass He WDs with masses below 0.25 Msun and surface gravities 5 < log g < 7, the so-called extremely low-mass helium white dwarfs (ELM WDs).
Today we know of the existence of more than 80 ELM WDs, with the majority of them in a binary system with a more massive CO WD. The increasing number of discovered ELM WDs reveals that they are formed in different environments, from the Galactic disk to open and globular clusters. ELM WDs are most likely the result of binary evolution as they cannot be formed from single stars within a Hubble time. The new wealth of data raises questions regarding the puzzling presence of metals in the atmospheres of young bloated ELM proto-WDs and the newly discovered pulsations in three ELM proto-WDs. The current available evolutionary models fail to explain such properties.
In this talk, I will present the latest theoretical efforts regarding the formation and evolution of ELM WDs.