Two years ago, we have reported a new class of materials that can be viewed as individual atomic planes pulled out of bulk crystals. Despite being only one atom thick and unprotected from the immediate environment, these two-dimensional (2D) materials were found to be stable under ambient conditions and exhibit remarkably high crystal quality, especially for the case of graphene (a single plane of carbon atoms). This is particularly surprising because such one-atom-thin materials previously were not known and presumed not to exist in the free state. Although it is nice to find a new material (and not just one but a whole class of them), it is even more important that the 2D materials turned out to exhibit truly exceptional properties and fascinating effects. For example, by attaching electrical contacts and measuring resistivity of graphene crystals, we have found that charge carriers in them move without scattering over thousands of inter-atomic distances, even under ambient conditions. This is despite the fact that the single layer of carbon atoms making up graphene is exposed from both sides to the immediate environment, i.e. air, moisture, etc. Another extraordinary feature of graphene is that its charge carriers behave as relativistic particles and, accordingly, are described by the (relativistic) Dirac equation rather than the (non-relativistic) Schrdinger equation. This allows access to the subtle and rich physics of quantum electrodynamics in a bench-top, condensed matter experiment. The latter has been proven in our recent experiments reporting several relativistic-like phenomena including two new types of the quantum Hall effects and quantum conductivity in the limit of a vanishing density of charge carriers.The fellowship would allow the applicant to dedicate all his time to the development of this research area, searching for new effects that are expected to be abundant in the new low-dimensional materials and finding their applications in various devices whose metallic, superconducting or magnetic properties could be controlled by gate voltage. Such devices are widely sought for many applications, and the odds are that 2D crystals can solve the problems that previously did not allow those many applications for the case of conventional materials.Since our publication describing 2D materials in Oct 2004, the research area is becoming increasingly active every month. For example, our paper on graphene has been acknowledged by Thomson Scientific Essential Science Indicators as one of the most cited recent papers in the field of Physics , and our second paper on the relativistic-like quantum Hall effect in graphene is predicted to be even more cited at the same stage. There have already been progress reports on graphene in Physics Today, Physics World and Materials Today, and first review articles are coming out soon in Nature Materials and Rev. Mod. Physics. Dozens of theoretical groups are now working on graphene and, although there are still very few experimental papers, the applicant is aware of at least two dozen of experimental groups who moved into the area (because most of them visited Manchester to learn about our fabrication of graphene crystals). Currently, the applicant is a clear leader in the field, and the fellowship is truly essential to maintain this position. The fellowship would also help to avert the danger that this UK-initiated research can fall under control of other large and well funded groups from the US, Europe and Japan. The fellowship seems to be most timely and cost-effective investment to maintain the strong UK presence in this newly-emerged research field.