The Gibbs sampler is for de novo motif discovery. Suppose we have a set of sequences, each containing a regulatory motif located in different locations of the sequences, but we do not know what the motif looks like or where it is located within each sequence. The Gibbs sampler will find such a motif if it is well represented in these sequences. If we have a set of yeast intron sequences, each containing a branchpoint site (BPS) somewhere, but we do not know what BPS looks like or where it is located along the intron sequence, the Gibbs sampler will find these BPSs. Another scenario involves the discovery of protein binding sites (e.g., transcription factor-binding sites) given a set of sequences from ChIP-Seq. Each of these sequences has a short sequence segment with affinity to a protein, but we do not know what the short sequence segment looks like or where it is located within the sequence. Gibbs sampler shines in discovering such protein-binding sites. This chapter breaks the black box of the Gibbs sampler and numerically illustrates each of its computational steps, including the site sampler (which assumes that each input sequence harbors a signal motif) and motif sampler (which is used when some sequences may contain multiple signal motifs and some none). The algorithm is used to find the highly degenerate BPS in mammalian introns.