Everybody is current in a world surrounded by computer. Computers determine our professional activity and penetrate increasingly deeper into our everyday life. Therein we also need increasingly refined c- puter technology. Sometimes we think that the next generation of c- puter will satisfy all our dreams, giving us hope that most of our urgent problems will be solved very soon. However, the future comes and il- sions dissipate. This phenomenon occurs and vanishes sporadically, and, possibly, is a fundamental law of our life. Experience shows that indeed 'systematically remaining' problems are mainly of a complex tech- logical nature (the creation of new generation of especially perfect - croschemes, elements of memory, etc. ). But let us note that amongst these problems there are always ones solved by our purely intellectual efforts alone. Progress in this direction does not require the invention of any 'superchip' or other similar elements. It is important to note that the results obtained in this way very often turn out to be more significant than the 'fruits' of relevant technological progress. The hierarchical asymptotic analytical-numerical methods can be - garded as results of such 'purely intellectual efforts'. Their application allows us to simplify essentially computer calculational procedures and, consequently, to reduce the calculational time required. It is obvious that this circumstance is very attractive to any computer user.

General Ideas, Concepts, Definitions, and other Preliminary Information.- Hierarchy and Hierarchical Systems.- Hierarchical Asymptotic Methods. General Ideas.- Hierarchical Systems with Fast Rotating Phases.- Hierarchical Systems with Fast Rotating Phases. Examples of Practical Applications.- Hierarchical Systems with Partial Derivatives. Method of Averaged Characteristics.- Example: Application of the Method of Averaged Characteristics in Nonlinear Theory of the Two-Stream Instability.- Hierarchical Systems with Partial Derivatives. Some other Asymptotic Methods.