Born at Basel in Switzerland, Muller was educated at the Federal Institute of Technology, Zurich, where he obtained his PhD in 1958. After working for a few years at the Batelle Institute in Geneva, he returned to Zurich (1963) to take up a post at the IBM Research Laboratory at Rüschlikon, where he has remained ever since.

In 1911 Kamerlingh-Onnes discovered the phenomenon of superconductivity. He found that a current passing through mercury at 4 K, that is four degrees above absolute zero, met with no resistance. To utilize this discovery fully the temperature at which materials became superconductive, the critical temperature (T _c), would have to be raised to some more economically accessible level. Yet 75 years' intensive research had raised the critical temperature no higher than 23.3 K for a niobium–germanium alloy. And to cool the alloy to this point requires bathing it in either expensive liquid helium (bp 4.2 K) or the cheaper but flammable liquid hydrogen (bp 20.3 K).

Muller first began to work on the problem in 1983. He ignored the usual candidates for a high critical temperature and turned instead to look at ceramic metal oxides. This was partly because his laboratory had worked with oxides of this kind for many years and had built up a considerable expertise in them. Also, he suspected, their lattice structure was of the right kind to allow superconductivity. In January 1986 Muller, working with his IBM colleague Georg Bednorz, found that a mixed lanthanum, barium, and copper oxide showed a change to superconducting behavior below 35 K (–238° C). Once the initial advance had been made, other physicists were quick to follow and to confirm and extend Muller's work.

The significance of Muller's discovery was recognized with unusual speed by the Nobel authorities when, in the following year, they awarded the Nobel Prize for physics jointly to Muller and Bednorz.