The noun engagement has 7 senses (first 5 from tagged texts)
1. (6) battle, conflict, fight, engagement -- (a hostile meeting of opposing military forces in the course of a war; "Grant won a decisive victory in the battle of Chickamauga"; "he lost his romantic ideas about war when he got into a real engagement")
2. (2) date, appointment, engagement -- (a meeting arranged in advance; "she asked how to avoid kissing at the end of a date")
3. (2) betrothal, troth, engagement -- (a mutual promise to marry)
4. (1) employment, engagement -- (the act of giving someone a job)
5. (1) engagement, booking -- (employment for performers or performing groups that lasts for a limited period of time; "the play had bookings throughout the summer")
6. engagement, mesh, meshing, interlocking -- (contact by fitting together; "the engagement of the clutch"; "the meshing of gears")
7. engagement, participation, involvement, involution -- (the act of sharing in the activities of a group; "the teacher tried to increase his students' engagement in class activities")