We give conditions under which limited-memory quasi-Newton methods with exact line searches will terminate in $n$ steps when minimizing $n$-dimensional quadratic functions. We show that although all Broyden family methods terminate in $n$ steps in their full-memory versions, only BFGS does so with limited-memory. Additionally, we show that full-memory Broyden family methods with exact line searches terminate in at most $n+p$ steps when $p$ matrix updates are skipped. We introduce new limited-memory BFGS variants and test them on nonquadratic minimization problems. (Also cross-referenced as UMIACS-TR-96-49)