The paper gives a historical survey of roundoff error analysis starting with a reappraisal of the classical paper by von Neumann and Goldstine; it is shown that their results compare less unfavorably with recent error analyses than is generally supposed. The conscious adoption of backward error analysis and floating-point computation has greatly simplified error analysis and exposed the importance of the control of “growth” in matrix computations based on equivalence and similarity transformations. This has led to the use of unitary transformations with guaranteed numerical stability.