Although numerous archival XMM-Newton observations existed towards the Small Magellanic Cloud (SMC) before 2009, only a fraction of the whole galaxy had been covered. Between May 2009 and March 2010, we carried out an XMM-Newton survey of the SMC, to ensure a complete coverage of both its bar and wing. Thirty-three observations of 30 different fields with a total exposure of about one Ms filled the previously missing parts. We systematically processed all available SMC data from the European Photon Imaging Camera. After rejecting observations with very high background, we included 53 archival and the 33 survey observations. We produced images in five different energy bands. We applied astrometric boresight corrections using secure identifications of X-ray sources and combined all the images to produce a mosaic covering the main body of the SMC. We present an overview of the XMM-Newton observations, describe their analysis, and summarise our first results, which will be presented in detail in follow-up papers. Here, we mainly focus on extended X-ray sources, such as supernova remnants (SNRs) and clusters of galaxies, that are seen in our X-ray images. Our XMM-Newton survey represents the deepest complete survey of the SMC in the 0.15-12.0keV X-ray band. We propose three new SNRs that have low surface brightnesses of a few 10^-14^erg/s/cm^2^/arcmin^2^ and large extents. In addition, several known remnants appear larger than previously measured at either X-rays or other wavelengths extending the size distribution of SMC SNRs to larger values.