  Electron captures on nuclei play an essential role for the dynamics of
several astrophysical objects. The capture rate can be derived in perturbation
theory where allowed nuclear transitions (Gamow-Teller transitions) dominate,
except at the higher temperatures achieved in core-collapse supernovae where
also forbidden transitions contribute significantly to the rates. There has
been decisive progress in recent years in measuring Gamow-Teller (GT) strength
distributions using novel experimental techniques based on charge-exchange
reactions. These measurements provide not only data for the GT distributions of
ground states for many relevant nuclei, but also serve as valuable constraints
for nuclear models which are needed to derive the capture rates for the many
nuclei, for which no data exist yet. In particular models are needed to
evaluate the stellar capture rates at finite temperatures, where the capture
can also occur on excited nuclear states.
  There has also been significant progress in recent years in the modelling of
stellar capture rates. This has been made possible by advances in nuclear
many-body models as well as in computer soft- and hardware. Specifically to
derive reliable capture rates for core-collapse supernovae a dedicated strategy
has been developed based on a hierarchy of nuclear models specifically adapted
to the abundant nuclei and astrophysically conditions present at the various
collapse conditions.
  This manuscript reviews the experimental and theoretical progress achieved
recently in deriving stellar electron capture rates. It also discusses the
impact these improved rates have on the various astrophysical objects.
(Abridged)
