  The onset of quartetting, i.e. alpha-particle condensation, in symmetric
nuclear matter is studied with the help of an in-medium modified four nucleon
equation. It is found that at very low density quartetting wins over pairing,
because of the strong binding of the alpha-particles. The critical temperature
can reach values up to around 6 MeV. Also the disappearance of alpha-particles
with increasing density, i.e. the Mott transition, is investigated. In finite
nuclei the Hoyle state, that is the 0_2^+ of 12C, is identified as an
"alpha-particle condensate" state. It is conjectured that such states also
exist in heavier n alpha-nuclei, like 16O, 20Ne, etc. For instance the 6-th 0^+
state of 16O at 15.1 MeV is identified from a theoretical analysis as being a
strong candidate for an alpha condensate state. Exploratory calculations are
performed for the density dependence of the alpha condensate fraction at zero
temperature to address the suppression of the four-particle condensate below
nuclear-matter density. Possible quartet condensation in other systems is
discussed briefly
