The number of cycles of length k that can be generated by q-ary n-stage feedback shift registers is studied. This problem is equivalent to finding the number of cycles of length k in the natural generalization, from binary to q-ary digits, of the so-called de Bruijn-Good graphs. The number of cycles of length k in the q-ary graph 
of order n is denoted by 
. Known results about 
are summarized and extensive new numerical data is presented. Lower and upper bounds on 
are derived showing that, for large k, virtually all q-ary cycles of length k are contained in 
for 
, but virtually none of these cycles is contained in 
for 
. More precisely, if 
denotes the total number of q-ary lenth-k cycles, then for any function f(k) that grows without bounds as 
(e.g. 
), the bounds obtained on 
are asymptotically tight in the sense that they imply 
for 
, and 
for 
, where 
denotes the integer part of the enclosed number. Finally, some approximations for 
are given that make the global behaviour of 
more transparent.
