uspavljivati

1. to put to sleep; to lull to sleep
   Mama svako veče uspavljuje bebu pesmom. — Every evening Mom sings the baby to sleep.
2. (of something boring) to make sleepy; to bore
   Ovaj profesor nas uspavljuje na svakom predavanju. — This professor puts us to sleep at every lecture.
3. to lull; to dull
   Lepe reči samo uspavljuju našu budnost. — Fine words only make us let our guard down.
4. (in medicine) to anaesthetize; to sedate
   Pre operacije pacijente uspavljuju opštom anestezijom. — Before surgery, patients are put under general anaesthesia.
5. (of animals, euphemism) to put to sleep; to euthanize
   Veterinar nerado uspavljuje stare pse. — The vet is reluctant to put old dogs to sleep.

To oversleep is usually expressed by the perfective uspavati se.