Accurate determination of stellar atmospheric parameters and elemental abundances is crucial for Galactic archaeology via large-scale spectroscopic surveys. In this paper, we estimate stellar atmospheric parameters -- effective temperature T sub( eff), surface gravity log g and metallicity Fe/H, absolute magnitudes M sub( V) and M sub( Ks), ...-element to metal (and iron) abundance ratio .../M (and .../Fe), as well as carbon and nitrogen abundances C/H and N/H from the Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) spectra with a multivariate regression method based on kernel-based principal component analysis, using stars in common with other surveys (Hipparcos, Kepler, Apache Point Observatory Galactic Evolution Experiment) as training data sets. Both internal and external examinations indicate that given a spectral signal-to-noise ratio (SNR) better than 50, our method is capable of delivering stellar parameters with a precision of ~100 K for T sub( eff), ~0.1 dex for log g, 0.3-0.4 mag for M sub( V) and M sub( Ks), 0.1 dex for Fe/H, C/H and N/H, and better than 0.05 dex for .../M (.../Fe). The results are satisfactory even for a spectral SNR of 20. The work presents first determinations of C/H and N/H abundances from a vast data set of LAMOST, and, to our knowledge, the first reported implementation of absolute magnitude estimation directly based on a vast data set of observed spectra. The derived stellar parameters for millions of stars from the LAMOST surveys will be publicly available in the form of value-added catalogues. (ProQuest: ... denotes formulae/symbols omitted.)