Context. The temperature and density structure of molecular cloud cores are the most important physical quantities that determine the course of the protostellar collapse and the properties of the... Show moreContext. The temperature and density structure of molecular cloud cores are the most important physical quantities that determine the course of the protostellar collapse and the properties of the stars they form. Nevertheless, density profiles often rely either on the simplifying assumption of isothermality or on observationally poorly constrained model temperature profiles. The instruments of the Herschel satellite provide us for the first time with both the spectral coverage and the spatial resolution that is needed to directly measure the dust temperature structure of nearby molecular cloud cores. Aims: With the aim of better constraining the initial physical conditions in molecular cloud cores at the onset of protostellar collapse, in particular of measuring their temperature structure, we initiated the guaranteed time key project (GTKP) ''The Earliest Phases of Star Formation'' (EPoS) with the Herschel satellite. This paper gives an overview of the low-mass sources in the EPoS project, the Herschel and complementary ground-based observations, our analysis method, and the initial results of the survey. Methods: We study the thermal dust emission of 12 previously well-characterized, isolated, nearby globules using FIR and submm continuum maps at up to eight wavelengths between 100 {$μ$}m and 1.2 mm. Our sample contains both globules with starless cores and embedded protostars at different early evolutionary stages. The dust emission maps are used to extract spatially resolved SEDs, which are then fit independently with modified blackbody curves to obtain line-of-sight-averaged dust temperature and column density maps. Results: We find that the thermal structure of all globules (mean mass 7 M$_{⊙}$) is dominated by external heating from the interstellar radiation field and moderate shielding by thin extended halos. All globules have warm outer envelopes (14-20 K) and colder dense interiors (8-12 K) with column densities of a few 10$^{22}$ cm$^{-2}$. The protostars embedded in some of the globules raise the local temperature of the dense cores only within radii out to about 5000 AU, but do not significantly affect the overall thermal balance of the globules. Five out of the six starless cores in the sample are gravitationally bound and approximately thermally stabilized. The starless core in CB 244 is found to be supercritical and is speculated to be on the verge of collapse. For the first time, we can now also include externally heated starless cores in the L$_{smm}$/L$_{bol}$ vs. T$_{bol}$ diagram and find that T$_{bol}$ {lt} 25 K seems to be a robust criterion to distinguish starless from protostellar cores, including those that only have an embedded very low-luminosity object. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.Partially based on observations carried out with the IRAM 30 m Telescope, with the Atacama Pathfinder Experiment (APEX), and with the James Clerk Maxwell Telescope (JCMT). IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). APEX is a collaboration between Max Planck Institut für Radioastronomie (MPIfR), Onsala Space Observatory (OSO), and the European Southern Observatory (ESO). The JCMT is operated by the Joint Astronomy Centre on behalf of the Particle Physics and Astronomy Research Council of the United Kingdom, the Netherlands Association for Scientific Research, and the National Research Council of Canada.Appendices A, B and C are available in electronic form at http://www.aanda.orgShow less
