  We introduce several classes of localizations (idempotent monads) on the
category of groups and study their properties and relations. The most
interesting class for us is the class of localizations which coincide with
their zero derived functors. We call them right exact (in the sense of Keune).
We prove that a right exact localization $L$ preserves the class of nilpotent
groups and that for a finite $p$-group $G$ the map $G\to LG$ is an epimorphism.
We also prove that some examples of localizations (Baumslag's $P$-localization
with respect to a set of primes $P,$ Bousfield's $HR$-localization, Levine's
localization, Levine-Cha's $\mathbb Z$-localization) are right exact. At the
end of the paper we discuss a conjecture of Farjoun about Nikolov-Segal maps
and prove a very special case of this conjecture.
