  The main achievements, current developments and prospects of molecular
studies in external galaxies are reviewed. They are put in the context of the
results of several decades of studies of molecules in local interstellar
medium, their chemistry and their importance for star formation. CO
observations have revealed the gross structure of molecular gas in galaxies.
Together with other molecules, they are among the best tracers of star
formation at galactic scales. Our knowledge about molecular abundances in
various local galactic environments is progressing. They trace physical
conditions and metallicity, and they are closely related to dust processes and
large aromatic molecules. Major recent developments include mega-masers, and
molecules in Active Galactic Nuclei; millimetre emission of molecules at very
high redshift; and infrared H2 emission as tracer of warm molecular gas, shocks
and photodissociation regions. The advent of sensitive giant interferometers
from the centimetre to sub-millimetre range, especially ALMA in the near future
in the mm/submm range, will open a new area for molecular studies in galaxies
and their use to trace star formation at all distances.
