An interactive system for symbolic computation, Maple was developed to perform hundreds of functions for use at all mathematical levels. This manual describes all of the functions that are available in the Maple library which is where the code for over 95% of Maple's mathematical knowledge and expertise resides. Through the library, the Maple system and the algorithms that it use can be dynamically extended and customized. Each description of the Maple library routines includes a short explanation of the function, the parameters that it uses, and additional details about the function. The design and implementation of the Maple system is an on-going project of the Symbolic Computation Group at the University of Waterloo. The Maple Symbolic Computation System is currently being used at over 1000 sites worldwide. An accompanying manual, the "Maple Language Reference Manual", which describes the Maple language and the Maple system itself is currently in preperation. This reference book on mathematics, physics/astronomy, engineering and computer science is intended for mathematicians, computer scientists and physicists.