Hox Gene Expression starts with the amazing discovery of the homeobox twenty-three years ago and follows the exciting path thereafter of a series of breakthroughs in Genetics, Development and Evolution. It deals with homeotic genes, their evolution, structure, normal and abnormal function. Researchers and graduate students in biology and medicine will benefit from this integrated overview of Hox gene activities.

This 121st IMA volume, entitled MATHEMATICAL MODELS FOR BIOLOGICAL PATTERN FORMATION is the first of a new series called FRONTIERS IN APPLICATION OF MATHEMATICS. The FRONTIERS volumes are motivated by IMA pro grams and workshops, but are specially planned and written to provide an entree to and assessment of exciting new areas for the application of mathematical tools and analysis. The emphasis in FRONTIERS volumes is on surveys, exposition and outlook, to attract more mathematicians and other scientists to the study of these areas and to focus efforts on the most important issues, rather than papers on the most recent research results aimed at an audience of specialists. The present volume of peer-reviewed papers grew out of the 1998-99 IMA program on "Mathematics in Biology," in particular the Fall 1998 em phasis on "Theoretical Problems in Developmental Biology and Immunol ogy." During that period there were two workshops on Pattern Formation and Morphogenesis, organized by Professors Murray, Maini and Othmer. James Murray was one of the principal organizers for the entire year pro gram. I am very grateful to James Murray for providing an introduction, and to Philip Maini and Hans Othmer for their excellent work in planning and preparing this first FRONTIERS volume. I also take this opportunity to thank the National Science Foundation, whose financial support of the IMA made the Mathematics in Biology pro gram possible.

Somite development is a unique and important topic; unique because somitic mesoderm is the first tissue to become segmented, important because the somites - once established - exert a profound influence on other seg mental structures which form later. Somite development is of interest at a number of levels. In one aspect, demarcation of a specific number of somites and size regulation, it is a particularly intriguing example of embryonic pattern formation, 'especially in vew of its possible relation to homoeobox-controlled segmentation in insects. At another level, somite development has long been studied by compara tive anatomists, but only recently has new light been thrown on such sub jects as segmentation of the head, proposed in the 18th century by Goethe and now a live issue for developmental biologists. Somit es are simple when they first appear, but very complex structures arise from them. These include the vertebrae and the axial and other muscles and consequently there is a wealth of morphogenetic problems to be explored. Sometimes their morphogenesis is disturbed, by genetic or environ mental factors, and there are many clinical conditions which arise as a result.

Following pioneering work by Harrison on amphibian limbs in the 1920s and by Saunders (1948) on the apical ridge in chick limbs, limb development became a classical model system for investigating such fundamental developmental issues as tissue interactions and induction, and the control of pattern formation. Earlier international conferences, at Grenoble 1972, Glasgow 1976,and Storrs, Connecticut 1982, reflected the interests and technology of their time. Grenoble was concerned with ectoderm-mesenchyme interaction, but by the time of the Glasgow meeting, the zone of polarizing activity (ZPA) and its role in control of patterning was the dominant theme. Storrs produced the first intimations that the ZPA could be mimicked by retinoic acid (RA), but the diversity of extracellular masrix ~olecules,particularly in skeletogenesis,was the main focus of attention. By 1990, the paradigms had again shifted. Originally, the planners of the ARW saw retinoic acid (as a possible morphogen controlling skeletal patterning), the variety of extracellular matrix components and their roles, and the developmental basis of limb evolution as the leading contemporary topics. However, as planning proceeded, it was clear that the new results emerging from the use of homeobox gene probes (first developed to investigate the genetic control of patterning of Drosophila embryos) to analyse the localised expression of "patterning genes" in limb buds would also be an important theme.