Important developments in one-dimensional real dynamics include substantial progress in our understanding of circle maps, quadratic maps of the interval, polynomial maps of an interval, unimodal and multimodal maps. Some of the highlights are the work by Herman and Yoccoz on circle maps, Swiatek-Graczyk and Lyubich on density of hyperbolicity for quadratic maps, Lyubich's work on stochasticity versus periodicity, as well as work by de Melo, van Strien, Shen, Avila and others. The area has been extremely successful but less progress has been made in higher dimension. The present program is mainly emphasizing the theory of two- dimensional maps where there are many open problems, but we may also consider dimension ≥ 3 and even infinite dimension (PDE:s). The subjects investigated include area preserving maps, dissipative maps, Anosov systems, quasi periodic skew products, hyperbolic skew products, maps of entropy zero, and piecewise isometries.