  For a set of integers $I$, we define a $q$-ary $I$-cycle to be a assignment
of the symbols 1 through $q$ to the integers modulo $q^n$ so that every word
appears on some translate of $I$. This definition generalizes that of de Bruijn
cycles, and opens up a multitude of questions. We address the existence of such
cycles, discuss ``reduced'' cycles (ones in which the all-zeroes string need
not appear), and provide general bounds on the shortest sequence which contains
all words on some translate of $I$. We also prove a variant on recent results
concerning decompositions of complete graphs into cycles and employ it to
resolve the case of $|I|=2$ completely.
