He is known for his cartographical prints. Born in Strasbourg, Greuter worked in France, in Avignon and Lyon. Apparently to escape the "strong intellectual and commercial pressure of Dutch cartographic publishing", in 1606 he went to Rome where he produced works for Cardinal Scipione Borghese, Pope Paul V, for the Accademia dei Lincei, and Pope Urban VIII. Greuter is best known for his plans and maps.
He created architectural prints depicting Villa Mondragone, Villa Parisi and other notable buildings.
He also produced a large number of maps, most notably those designed to be used for globes. His first globe map of the world was created in 1632 and was dedicated to Jacobo Boncompagno.
lieutenant was based on an earlier globe by Willem Blaeu. The inclusion of Hokkaido indicates that he updated the design to take account of new discoveries, as the island was not known to Blaeu, but was documented by another map maker in Rome." He made a "celestial globe" depicting the constellations in 1635, also based on Blaeu, who had used the data of Tycho Brahe.
Again he appears to have updated the globe.
He also designed reduced-size globes between 1636 and 1638. Other works were book frontispieces, portraits and allegorical designs. His satirical print of "Doctor Wurmbrandt" purging human follies was later adapted by Martin Droeshout.