This book probes mathematical issues that arise in modeling interest rate term structure, by casting the interest rate models as stochastic evolution equations in infinite dimensions. The book is comprised of three parts. Part I is a crash course on interest rates, including a statistical analysis of the data and an introduction to some popular interest rate models. Part II is a self-contained introduction to infinite dimensional stochastic analysis, including SDE in Hilbert spaces and Malliavin calculus. Part III presents recent results in interest rate theory, including finite dimensional realizations of HJM models, generalized bond portfolios, and the ergodicity of HJM models.