We generalize the notion of a de Bruijn sequence to a "multi de Bruijn sequence": a cyclic or linear sequence that contains every k-mer over an alphabet of size q exactly m times. For example, over the binary alphabet {0, 1}, the cyclic sequence (00010111) and the linear sequence 000101110 each contain two instances of each 2-mer 00, 01, 10, 11. We derive formulas for the number of such sequences. The formulas and derivation generalize classical de Bruijn sequences (the case m = 1). We also determine the number of multisets of aperiodic cyclic sequences containing every k-mer exactly m times; for example, the pair of cyclic sequences (00011)(011) contains two instances of each 2-mer listed above. This uses an extension of the Burrows-Wheeler Transform due to Mantaci et al, and generalizes a result by Higgins for the case m = 1.LinearLinear, starts with k-mer y L y (m, q, k) W (m, q, k)/q k Linearized, starts with y LC y (m, q, k) W (m, q, k)/q k Cyclic, order d C (d) (m, q, k)