In this thesis I present the spectroscopic analysis of a large sample of L and T dwarfs, in order to constrain the sub-stellar initial mass function and formation history. The main points I tried to address are the development of a better spectral type to distance calibration and of a better spectral type to effective temperature calibration, and the identification of a statistically complete sample of brown dwarf to be used to measure the luminosity function, and therefore to constrain the initial mass function and formation history. To achieve the first goal I conducted the spectroscopic follow-up of brown dwarfs from the PARallaxes of Southern Extremely Cool objects (PARSEC) program. This is a large astrometric campaign to measure the parallaxes and proper motions of 120 L and T dwarfs in the southern hemisphere. I combined the astrometric results with the near infra-red spectra I obtained using the Ohio State Infra-Red Imager/Spectrometer (OSIRIS) on the Southern Astrophysical Research telescope (SOAR). That allowed me to investigate the nature of some unresolved binaries and common proper motion companion in the sample, as well as sub-dwarfs candidates, and potential members of young moving groups. Combining the spectra with the astrometric information and the available photometry I derived the bolometric luminosity and effective temperature for the targets, and determined a new polynomial conversion between spectral type and effective temperature of a brown dwarfs. This is a fundamental step to compare the results of empirical observations to numerical simulations of the sub-stellar luminosity function. Once refined the type to temperature calibration, I measured the luminosity function. In order to do so my collaborators and I have selected a sample of 250 brown dwarfs candidates from the United Kingdom Deep Infra-red Sky Survey (UKIDSS) Large Area Survey (LAS) and followed them up with the echelle spectrograph X-shooter on the Very Large Telescope. I present in this thesis the results of the observations of 196 of the brown dwarfs candidates. Using the X-shooter spectra I determined their spectral types, and I identified a number of unresolved binary candidates and peculiar objects. One of the peculiar objects in the sample, ULAS J222711 004547, turned out to be the reddest brown dwarf observed so far, and I therefore proceeded to analyse further its spectrum. Applying a de-reddening technique to its spectrum suggests that the most likely reason for its redness is an excess of dust in its photosphere, and that can account for the differences seen between objects of similar spectral type. By comparing the results of the spectroscopic campaign to numerical simulations, I found that it is currently impossible to constrain robustly the initial mass function and formation history of sub-stellar objects, because of our limited knowledge of the binary fraction among brown dwarfs. The sample of binary candidates identified in this thesis can be used to place a better constraint on the binary fraction, but in order to do that the candidates need to be followed-up via high resolution imaging or radial velocity monitoring to confirm their binary nature.