Biology is encoded in molecular sequences: deciphering this encoding remains a grand scientific challenge. Functional regions of DNA, RNA, and protein sequences often exhibit characteristic but subtle motifs; thus, computational discovery of motifs in sequences is a fundamental and much-studied problem. However, most current algorithms do not allow for insertions or deletions (indels) within motifs, and the few that do have other limitations. We present a method, GLAM2 (Gapped Local Alignment of Motifs), for discovering motifs allowing indels in a fully general manner, and a companion method GLAM2SCAN for searching sequence databases using such motifs. glam2 is a generalization of the gapless Gibbs sampling algorithm. It re-discovers variable-width protein motifs from the PROSITE database significantly more accurately than the alternative methods PRATT and SAM-T2K. Furthermore, it usefully refines protein motifs from the ELM database: in some cases, the refined motifs make orders of magnitude fewer overpredictions than the original ELM regular expressions. GLAM2 performs respectably on the BAliBASE multiple alignment benchmark, and may be superior to leading multiple alignment methods for “motif-like” alignments with N- and C-terminal extensions. Finally, we demonstrate the use of GLAM2 to discover protein kinase substrate motifs and a gapped DNA motif for the LIM-only transcriptional regulatory complex: using GLAM2SCAN, we identify promising targets for the latter. GLAM2 is especially promising for short protein motifs, and it should improve our ability to identify the protein cleavage sites, interaction sites, post-translational modification attachment sites, etc., that underlie much of biology. It may be equally useful for arbitrarily gapped motifs in DNA and RNA, although fewer examples of such motifs are known at present. GLAM2 is public domain software, available for download at http://bioinformatics.org.au/glam2. In recent decades, scientists have extracted genetic sequences—DNA, RNA, and protein sequences—from numerous organisms. These sequences hold the information for the construction and functioning of these organisms, but as yet we are mostly unable to read them. It has long been known that these sequences contain many kinds of “motifs”, i.e. re-occurring patterns, associated with specific biological functions. Thus, much research has been devoted to computer algorithms for automatically discovering subtle, recurring motifs in sequences. However, previous algorithms search for rigid motifs whose instances vary only by substitutions, and not by insertions or deletions. Real motifs are flexible, and do vary by insertions and deletions. This study describes a new computer algorithm for discovering motifs, which allows for arbitrary insertions and deletions. This algorithm can discover real, flexible motifs, and should be able to help us determine the functions of many biological molecules.