  We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and
B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step
linear fractional recurrences. The first Theorem in this paper shows that for
each k there are k-step linear fractional recurrences which are periodic of
period 4k. Among this class of recurrences, there is also the so-called Lyness
process, which has the form $A(z)/B(z) = (a +z_{n+1} + z_{n+2} + ... +
z_{n+k-1})/z_n$. The second Theorem shows that the Lyness process has quadratic
degree growth. The Lyness process is integrable, and we discuss its known
integrals.
