For a long time there have been two kinds of mathematical computation: symbolic and numerical. Symbolic computing manipulates algebraic expressions exactly, but it is unworkable for many applications since the space and time requirements grow combinatorially. Numerical computing avoids the combinatorial explosion by rounding to 16 digits at each step, but it works just with individual numbers, not algebraic expressions. This talk will describe a new kind of computing that combines the feel of symbolics with the speed of numerics. The starting idea was to represent functions by Chebyshev expansions whose length is determined adaptively to maintain an accuracy of close to machine precision. The chebfun system is implemented in object-oriented Matlab, with familiar vector operations such as sum and diff being overloaded to analogues for functions such as integration and differentiation. But by now, the capabilities of chebfun have developed far beyond what this short description may suggest - including, for example, the high-precision automatic solution of linear and nonlinear differential equations by executing "backslash". Chebfun is a joint project with Rodrigo Platte, Nick Hale, Toby Driscoll, and Ricardo Pachon, and a crucial contribution was made by Folkmar Bornemann.