Combinators are shown to provide a very suitable basis for implementations of functional and symbolic computation in computer architecture. A powerful combinator reduction system is developed which meets programmers and machine requirements for (i) efficiency of representation and execution of symbolic algorithms, and (ii) availability of algebraic manipulation needed to analyse symbolic computations. An algebraic model is constructed to provide rigorous semantics for the system. The reduction language of the system aims at exposing efficient flows of data and fine-grain parallelism, and a computer architecture, which is proposed to run the system, utilizes both sequential and parallel processing modes in order to achieve maximum efficiency of symbolic computation. Finally, an implementation of the interpreter and functional simulator for the architecture is described.