• From 1772 to 1776, he received a pension that enabled him to study in Rome.
• There, he was able to work with Anton Raphael Mengs.
• From 1785, he was a councilor and Professor at the Vienna Academy's elementary drawing school; a position he would hold until 1817.
• His numerous well known students include Karl Agricola, Johann Scheffer von Leonhardshoff, Moritz Michael Daffinger, Ferdinand Georg Waldmüller, Karl Ruß, Wilhelm August Rieder, Kilian Ponheimer, Peter Fendi, Eustație Altini,
Johann Baptist von Lampi, Friedrich von Amerling and Johann Michael Sattler, who would later become Maurer's biographer.
• He died in Vienna, and a street in his birthplace of Bonn has been named after him.