We present intermediate- and low-resolution optical spectra around Hα and Li I lambda 6708 Åfor a sample of 25 low mass stars and 2 brown dwarfs with confirmed membership in the pre-main sequence stellar sigma Orionis cluster. Our observations are intended to investigate the age of the cluster. The spectral types derived for our target sample are found to be in the range K6-M8.5, which corresponds to a mass interval of roughly 1.2-0.02 Msun on the basis of state-of-the-art evolutionary models. Radial velocities (except for one object) are found to be consistent with membership in the Orion complex. All cluster members show considerable Hα emission and the Li I resonance doublet in absorption, which is typical of very young ages. We find that our pseudo-equivalent widths of Hα and Li I (measured relative to the observed local pseudo-continuum formed by molecular absorptions) appear rather dispersed (and intense in the case of Hα ) for objects cooler than M3.5 spectral class, occurring at the approximate mass where low mass stars are expected to become fully convective. The least massive brown dwarf in our sample, S Ori 45 (M8.5, ~ 0.02 Msun), displays variable Hα emission and a radial velocity that differs from the cluster mean velocity. Tentative detection of forbidden lines in emission indicates that this brown dwarf may be accreting mass from a surrounding disk. We also present recent computations of Li I lambda 6708 Åcurves of growth for low gravities and for the temperature interval (about 4000-2600 K) of our sample. The comparison of our observations to these computations allows us to infer that no lithium depletion has yet taken place in sigma Orionis, and that the observed pseudo-equivalent widths are consistent with a cluster initial lithium abundance close to the cosmic value. Hence, the upper limit to the sigma Orionis cluster age can be set at 8 Myr, with a most likely value around 2-4 Myr. Based on observations made with the following telescopes: 3.5-m telescope at the Spanish-German Calar Alto Observatory (Spain) operated by the Max-Planck-Institut für Astronomie in Heidelberg (Germany); 2.5-m Isaac Newton telescope operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de Los Muchachos of the Instituto de Astrofísica de Canarias; 2.1-m Otto Struve telescope at McDonald Observatory (USA); and the 10-m Keck II telescope of the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration (the Observatory was made possible by the generous financial support of the W. M. Keck Foundation).