Information theory celebrates its 50-th birthday. Although it is a mature area of research by any standard, new challenges arise due to new applications and new theoretical developments (e.g., there is a resurgence of interest in source coding in multimedia applications). In the 1997 Shannon Lecture Jacob Ziv presented compelling arguments for "backing off" to a certain degree from the (first-order) asymptotic analysis of information systems in order to predict the behavior of real systems where we always face finite (and often small) lengths (of sequences, files, codes, etc.). One way of overcoming these difficulties is to increase the accuracy of asymptotic analysis by replacing first-order analyses (e.g., a leading term of the average code length) by full asymptotic expansions and more accurate analyses (e.g., large deviations, central limit laws). We propose to accomplish this goal by exploring problems of information theory by analytic methods, that is, those in which complex analysis plays a pivotal role. Among others we propose research on lossless Lempel-Ziv schemes, lossy extension of Lempel-Ziv schemes (based on approximate pattern matching), context quantization (which aims at extending context-tree weighting to lossy environment), prediction schemes based on pattern matching, and hierarchy of redundancy rates. Analytic methods discussed here are: asymptotic analysis of functional-differential equations, poissonization and depoissonization, and complex asymptotics. This program constitutes a basis of what we call analytic information theory: It deals with problems of information theory that can be solved by analytic techniques which "are extremely powerful and when they apply, they often yield estimates of unparalleled precision" (Odlyzko 1995).