jats:pWe present full-sky maps of the cosmic microwave background (CMB) and polarized synchrotron and thermal dust emission, derived from the third set ofjats:italicPlanck</jats:italic>frequency maps. These products have significantly lower contamination from instrumental systematic effects than previous versions. The methodologies used to derive these maps follow closely those described in earlier papers, adopting four methods (jats:monospaceCommander</jats:monospace>,jats:monospaceNILC</jats:monospace>,jats:monospaceSEVEM</jats:monospace>, andjats:monospaceSMICA</jats:monospace>) to extract the CMB component, as well as three methods (jats:monospaceCommander</jats:monospace>,jats:monospaceGNILC</jats:monospace>, andjats:monospaceSMICA</jats:monospace>) to extract astrophysical components. Our revised CMB temperature maps agree with corresponding products in thejats:italicPlanck</jats:italic>2015 delivery, whereas the polarization maps exhibit significantly lower large-scale power, reflecting the improved data processing described in companion papers; however, the noise properties of the resulting data products are complicated, and the best available end-to-end simulations exhibit relative biases with respect to the data at the few percent level. Using these maps, we are for the first time able to fit the spectral index of thermal dust independently over 3° regions. We derive a conservative estimate of the mean spectral index of polarized thermal dust emission ofjats:italicβ</jats:italic>jats:subd</jats:sub> = 1.55  ±  0.05, where the uncertainty marginalizes both over all known systematic uncertainties and different estimation techniques. For polarized synchrotron emission, we find a mean spectral index ofjats:italicβ</jats:italic>jats:subs</jats:sub> = −3.1  ±  0.1, consistent with previously reported measurements. We note that the current data processing does not allow for construction of unbiased single-bolometer maps, and this limits our ability to extract CO emission and correlated components. The foreground results for intensity derived in this paper therefore do not supersede correspondingjats:italicPlanck</jats:italic>2015 products. For polarization the new results supersede the corresponding 2015 products in all respects.</jats:p>