We first study the problem of free dendritic growth from a pure substance. Dendritic growth from a binary mixture will be studied in Chap. 8 Typical free dendritic growth is shown in Fig. 1.6 In general, dendritic growth, starting from a tiny seed, proceeds via a very complex dynamic process until, at a later stage of evolution, a permanent pattern is displayed. The details of pattern formation during the entire evolution will obviously be affected by many factors, in particular by the details of the initial conditions. To describe and predict the whole history of growth is very difficult and not meaningful.