occupy or attract (someone's interest or attention)

he ploughed on, trying to outline his plans and engage Sutton's attention

I told him I was otherwise engaged

participate or become involved in

organizations engage in a variety of activities

some are actively engaged in crime

arrange to employ or hire (someone)

he was engaged as a trainee copywriter

(with reference to a part of a machine or engine) move into position so as to come into operation

the clutch will not engage

the driver engaged the gears and pulled out into the road

(of fencers or swordsmen) bring (weapons) together preparatory to fighting

(of a writer or artist) committed to a particular aim or cause

he was determined that his writing and thinking should be engagé