One of the most important applications of LOFAR will be to carry out large-sky surveys. Such surveys are well suited to the characteristics of LOFAR and have been designated as one of the key projects that have driven LOFAR since its inception. Such deep LOFAR surveys of the accessible sky at several frequencies will provide unique catalogues of radio sources for investigating several fundamental areas of astrophysics, including the formation of massive black holes, galaxies and clusters of galaxies. Because the LOFAR surveys will probe unexplored parameter space, it is likely that they will discover new phenomena. The following known classes of extragalactic objects are of particular interest: - Most Distant Radio Galaxies. The most efficient method for finding extremely distant radio galaxies uses an empirical correlation between radio spectral steepness and distance. Using this method, LOFAR will efficiently pick out radio galaxies at larger distances than possible with present radio telescopes. Detecting such objects before the epoch of reionisation would be particularly important. Not only would this yield important constraints on how and when massive black holes are formed, but this would also allow detailed studies of the ISM at these high redshifts through redshifted 21 cm absorption studies. Study of these distant radio galaxies at other wavelengths will provide information about the formation of massive galaxies and links between nuclear activity and star formation. Since distant radio galaxies pinpoint proto-clusters, studying the environments of these distant galaxies will constrain the formation of clusters at the earliest epochs. - Diffuse Emission in Galaxy Clusters. Clusters often contain diffuse radio sources that are shaped by the dynamics of the gas in which they are embedded. Their large extent, low surface brightness, and steep spectra makes these cluster sources difficult to study with conventional facilities, such as Westerbork and the VLA. Because of their steep spectra, LOFAR will be able to detect and study such radio sources in the many tens of thousands of clusters up to redshifts of two that will be detected using the XMM-Newton X-ray telescope, the Planck satellite, and the Sloan Digital Sky Survey. Such studies will be very relevant for (i) understanding the effects on the dynamics of cluster gas due to shock waves produced by cluster mergers, (ii) determining the origin of cluster magnetic fields, and (iii) measuring the occurrence and characteristics of diffuse cluster radio sources as a function of redshift with the aim of constraining physical models for the origin of these sources. - Star-forming Galaxies. Because of the large fields of view, surveys at the higher LOFAR frequencies will detect unprecedented numbers of star-forming galaxies with star formation rates of tens of solar masses per year at an epoch at which the bulk of galaxy formation is believed to occur. Since the ratio of radio flux to sub-mm flux is a sensitive redshift indicator, LOFAR surveys, in combination with data from new far-IR and millimeter facilities such as the SCUBA-2 sub-mm imaging array, the Spitzer and Herschel satellites and the ALMA (sub)mm array, will also provide distances and thus facilitate a census of the cosmic star formation history, unhindered by the effects of dust obscuration. For nearby galaxies, LOFAR will enable spectral mapping at low frequencies, thereby providing unique information about absorption and delineating the spatial distribution of the warm ISM and galactic magnetic fields. The survey key project will be optimised for obtaining large samples of the various categories of extragalactic radio emitting objects listed above. The provisional plan is to survey the entire accessible sky at a number of 'key frequencies', 30, 75, 120, and 200 MHz (initially). An additional key frequency of 15 MHz will be added after the 10-30 MHz optimized LOFAR upgrade has been implemented. The surveys will be carried out over several epochs, enabling variable sources to be recognised.