We demonstrate experimentally, for the first time, a new form of lensless microscopy. The image we obtain contains the entire wavefunction emanating from the sample. Large scale, quantitative phase information can be measured, unlike in conventional (Zernike) methods. For light optical experiments, we can dispense with expensive high-quality lenses and the very large working distances available would allow remote monitoring of e.g., environmental cells without compromising resolution. In short wavelength microscopy (X-rays and electrons), where lens components are of very limited numerical aperture, the technique has revolutionary implications: objects of any lateral size or shape can be used and, for transmission electron imaging, resolution down to the scale of the wavelength is likely to be limited only by the presence of atomic vibrations.