At the beginning of the 20th century, scientists discovered how to measure the size and spacing of atoms using a technique called diffraction, which led to a revolution in the understanding of chemistry, biology and solid-state physics. X-rays and electrons behave like waves, but with a wavelength which is much smaller than the spacing between the atoms of a solid. These waves scatter and interfere with one another, producing strong beams coming out of the object at particular angles. By measuring these angles, and knowing the wavelength of the waves, the separation of atoms could be calculated. It was using this method that Watson and Crick determined the structure of DNA in the 1950s. However, diffraction is only useful if the object is a regular lattice structure. In order to look at more complicated atomic structures, scientists have relied on electron or X-ray microscopes. In a standard microscope, a lens is used to produce a magnified image, but the method still relies on the waves that make up the radiation (light, electrons or X-rays) interfering with one another to build up the image. With light, this is experimentally easy, but with very-short wavelength radiation (a fraction of an atomic diameter), the tiniest error in the lens or the experimental apparatus makes the waves interfere incorrectly, ruining the image. For this reason, a typical electron or X-ray microscope image is about one hundred times more blurred than the theoretical limit defined by the wavelength.In this project, we aim to unify the strengths of the above apparently very different techniques to get the best-ever pictures of individual atoms in any structure (which is not necessarily crystalline). Our approach is to use a conventional (relatively bad) X-ray or electron lens to form a patch of moderately-focussed illumination (like burning a hole in a piece of paper with the sun's rays through a magnifying glass). In fact, we do not need a lens at all! Just a moveable aperture put in front of the object of interest will suffice. We then record the intensity of the diffraction pattern which emerges from the other side of the object on a good-quality high-resolution detector, for several positions of the illuminating beam. This data does not look anything like the object, but we have worked out a way of calculating a very good image of the object by a process called 'phase-retrieval'. To make an image of an object we have to know what's called the relative phase (the different arrival times) of the waves that get scattered from it. In diffraction, this information is lost, although some of it is preserved (badly) by a lens. Our data is a complex mixture of diffraction and image data, but the key innovation in this project is that we can use a computer to calculate the phase of the very high resolution data which could never be seen by the lens alone. Other workers in the United States have demonstrated very limited versions of this new approach, but we have a much more sophisticated computational method which eliminates essentially all earlier restrictions.The new method, which has received patent protection, could be implemented on existing electron or X-ray microscopes, greatly enhancing their imaging capability. It is even possible to contemplate a solid-state optical microscope, built into a single chip with no optical elements at all. All the weakness and difficulties and costs of lenses would be replaced by a combination of good quality detectors and computers. Our ultimate aim is to be able to image in 3D directly (using X-rays or electrons) any molecular structure, although this will require a great deal of research. The work put forward in this proposal will build the Basic Technology foundations of this new approach to the ultimate microscope.