Postdoctoral Opportunities in Cambridge
The Department of Pure Mathematics and Mathematical Statistics (DPMMS) at Cambridge has a large number of faculty, postdocs and graduate students, with active research groups across the spectrum of mathematics. Strengths include:
 algebra (group theory, representation theory);
 algebraic geometry (higherdimensional algebraic geometry, algebraic cycles, abelian varieties, mirror symmetry, geometric aspects of representation theory);
 analysis (geometric analysis, PDEs, mathematical relativity, discrete analysis);
 category theory (logic, proof theory, topos theory, higherdimensional category theory)
 combinatorics (random structures, combinatorial number theory, Ramsey theory, graph theory)
 differential geometry (dynamics, lowdimensional topology and knot theory, gauge theory, Riemannian geometry, symplectic topology);
 number theory (algebraic number theory, Iwasawa theory, computational number theory);
 probability and statistics (including financial mathematics, operational research, stochastic methods, transport and networking problems).
More detailed information can be found on other departmental webpages. If you have interests in these areas, particularly in the areas of expertise of departmental members, consider coming to Cambridge for a postdoc. (Related research groups at Imperial College, London and at Oxford are close, and there are close links  including shared participation in some seminars  with these departments, which adds to the richness of the local mathematical community.) There are various kinds of postdoc available, suitable for people at different levels, and requiring different application procedures. Some more information on each is available below.
 HERCHELSMITH POSTDOCTORAL RESEARCH FELLOWSHIPS:
A Fellowship in Mathematics is advertised as part of a University wide scheme. We expect to appoint one or more positions each year. The postdocs are tenable for 3 years and are on a competitive salary scale. The positions carry no teaching obligations, and are purely intended for research, although opportunities to give Masterslevel or graduatelevel courses are available. Applications from all areas of pure mathematics will be considered. Applicants should be towards the end of their PhD or within the first 23 years of postdoctoral research. Further particulars will be given on the departmental vacancies page when applications are open, which is typically around late October. 
COLLEGE RESEARCH FELLOWSHIPS:
These are prestigious postdocs, funded directly by the Cambridge Colleges. Holders of the positions are fully integrated into DPMMS, with office space in the department as well as their college. The precise details of these positions vary from college to college, and their availability varies from year to year. Typically, colleges offer up to 4 positions, for a tenure of 3 or 4 years. All positions are research positions, with few or minimal teaching obligations; the positions are associated with membership of a college, and typically come with accommodation and other perquisites. In some but not all cases there may be criteria of eligibility (e.g. citizen of an EU member state, or graduand of a UK university, etc). All such positions are advertised in the Cambridge University Reporter: www.admin.cam.ac.uk/reporter/current/weekly/; check regularly to see if there are openings in Mathematics. Note that although these positions are competitive across subjects, mathematicians are often successful: in a typical year there are 3 or more pure mathematicians appointed to such positions from the various colleges. For a further view see Professor Körner's unofficial
Guide for Mathematicians Seeking a Cambridge Research Fellowship

1851 ROYAL COMMISSION RESEARCH FELLOWSHIPS:
2year postdoctoral fellowships in mathematics and biological/physical sciences, which can be held at any UK university. Applicants should recently have completed or be about to complete a PhD: there are strict eligibility criteria (UK or Commonwealth national, or citizen of Ireland or Pakistan). Further information and application deadlines can be found here: www.royalcommission1851.org.uk/res_fellow.html 
MARIECURIE POSTDOCTORAL FELLOWSHIPS:
2 year researchonly postdoctoral positions, funded by the EU (to a generous salary), available to EU nationals to work in a country which is neither the country of their PhD nor their home country. Open calls for applications  which are often but not always present  can be found with some effort via the website cordis.europa.eu/fp7/home_en.html; follow links for Individual Researchers. MarieCurie grant applications are made jointly with the department, and are directly linked to the research of a specific member of faculty. The first stage in applying is to make contact with such a potential supervisor and discuss possible projects and establish whether or not there is support for and scope for a plausibly successful application. 
ROYAL SOCIETY UNIVERSITY RESEARCH FELLOWSHIPS:
Unique 8 year researchonly fellowships open to EU nationals. These are suitable for outstanding young mathematicians who have completed their PhD and already done one postdoc. Applications must have emphatic support from DPMMS; if you are interested in applying, contact a member of the department in a related subject. Further information is available from http://royalsociety.org/funding.asp?id=1121 The deadline is typically in very early November of the preceeding year. Results are only available in April, but who else is going to give you 8 years money with no obligations? 
EPSRC GRANTS:
Faculty members at Cambridge can write grants to the Engineering and Physical Sciences Research Council (UK government) for grants specifically to hire some particular postdoc for some particular research collaboration. These are quite often successful, although irksomely one doesn't tend to know if a grant application has been successful at the time one most wants to (i.e. when other offers come in). However, start dates are usually very flexible once the grant is awarded. If you're keen to come to Cambridge (especially if not an EU citizen, so various things above are not appropriate), then get in touch with someone in your field and see if this is a sensible option.
POSSIBILITIES NEARBY:
Imperial College London is only an hour from Cambridge, also has outstanding research groups in various fields (including the geometry group led by Simon Donaldson), and we often travel to and from one another's seminars. Oxford is slightly further but by no means inaccessible. Why not try applying there too? More information can be found on the websites http://www3.imperial.ac.uk/employment, and www.maths.ox.ac.uk/. Example opportunities include:  Chapman Fellowships at Imperial
 Colleges at the University of Oxford also offer Junior Research Fellowships, details are usually available via the Gazette www.ox.ac.uk/gazette/.