Linear programming has been a fundamental topic in the development of the computational sciences. The subject has its origins in the early work of L.B.J. Fourier on solving systems of linear inequalities, dating back to the 1820's. More recently, a healthy competition between the simplex and interior point methods has led to rapid improvements in the technologies of linear programming. This combined with remarkable advances in computing hardware and software have brought linear programming tools to the desktop, in a variety of application software for decision support. Linear programming has provided a fertile ground for the development of various algorithmic paradigms. Diverse topics such as symbolic computation, numerical analysis, computational complexity, computational geometry, combinatorial optimization, and randomized algorithms all have some linear programming connection. This chapter reviews this universal role played by linear programming in the science of algorithms.