prespavati

1. to stay the night; to sleep over
   Ako je kasno, možeš da prespavaš kod nas. — If it gets late, you can stay the night at our place.
   Prespavali smo u jednom motelu pored puta. — We spent the night at a roadside motel.
2. (+ acc.) to oversleep and miss; to sleep through
   Prespavao sam prvi čas i dobio neopravdan. — I slept through first period and got an unexcused absence.
   Prespavala je ceo film. — She slept through the whole film.
3. (of a decision, problem) to sleep on it
   Prespavaj to, pa mi sutra javi. — Sleep on it and let me know tomorrow.
4. (of a period of time) to spend a period asleep; to sleep away
   Prespavao je celo popodne. — He slept the whole afternoon away.

Unlike Russian "переспать", it has no sexual meaning; that is expressed by "spavati s kim".