Methods of computer algebra become more familiar to a wide audience of theoretical mathematicians and physicists. The environment of computer algebra system leads to a greater acceptance of computer instruments in the mathematical research. Methods of symbolic manipulation provided by computer algebra systems in combination with high-power number crunching abilities of traditional hardware and software open the way to truly large scale computations often needed by mathematicians and physicists. This chapter describes new low complexity (both operational and logical) methods and algorithms for computations of solutions of differential equations and their efficient evaluation, and of solution of algebraic equations. These computations are often basic in applied problems, but even more so in a variety of problems of number theory and algebraic geometry. Power series manipulations over various rings and fields are one of the most important features of advanced computer algebra systems.