Our understanding of the processes through which species adapt to one another is largely predicated upon the assumption that interactions between species depend on only single traits. However, real interactions, including those with implications for human health and agriculture, inevitably involve multiple traits. The primary reason multiple traits have not been integrated into existing theory is the sheer difficulty of developing and analyzing appropriate mathematical models. The proposed work will overcome existing hurdles by developing powerful new mathematical tools that allow a general theory of multivariate coevolution to be formulated. These tools will be made widely available to the scientific community through presentations at annual meetings and publication of electronic Mathematica appendices. In addition to significantly advancing our understanding of how species adapt to one another, the proposed work will promote undergraduate and graduate training at the interface of biology and mathematics. A team of students recruited into the University of Idaho's Undergraduates in Biological and Mathematical Sciences program will be trained in basic computational and mathematical approaches through work on this project. Graduate training will be accomplished by immersing a student into the proposed research, resulting in intensive training in advanced mathematical and computational approaches to biological problems.