We carried out a near-infrared radial velocity search for Jupiter-mass planets around 36 late M dwarfs. This survey was the first of its kind undertaken to monitor radial velocity variability of these faint dwarfs. For this unique survey we employed the 10-m Keck II on Mauna Kea in Hawaii. With a resolution of 20,000 on the near-infrared spectrograph, NIRSPEC, we monitored these stars over four epochs in 2007. In addition to the measurement of relative radial velocity we established physical properties of these stars. The physical properties of M dwarfs we determined included the identification of neutral atomic lines, the measurement of pseudo-equivalent widths, masses, surface gravity, effective temperature, absolute radial velocities, rotational velocities and rotation periods. The identification of neutral atomic lines was carried out using the Vienna Atomic line Database. We were able to confirm these lines that were previously identified. We also found that some of the lines observed in the K-type stars, such as Mg I though weak, still persist in late M dwarfs. Using the measurement of pseudo-equivalent widths (p-EW) of 13 neutral atomic lines, we have established relations between p-EW and spectral type. Such relations serve as a tool in determining the spectral type of an unknown dwarf star by means of measuring its p-EW. We employed the mass-luminosity relation to compute the masses of M dwarfs. Our calculations indicate these dwarfs to be in the range of 0.1 to 0.07 solar masses. This suggests that some of the late M dwarfs appear to be in the Brown dwarf regime. Assuming their radii of 0.1 solar radii, we calculated their surface gravity. The mean surface gravity is, log g = 5.38. Finally their effective temperature was determined by using the spectral-type iii temperature relationship. Our calculations show effective temperatures in the range of 3000 2300 K. Comparison of these values with models in literature show a good agreement. The absolute radial and rotational velocities of our targets were also calculated. Values of rotational velocities indicate that M dwarfs are, in general, slow rotators. Using our result and that from literature, we extended our study of rotational velocities to L dwarfs. Our observations show an increase in rotational velocities from late M to L dwarfs. We also find that the mean periods of M dwarfs are less than 10 hours. In order to improve our precision in measuring relative radial velocity (RV), we employed the use of deconvolution method. With this method we were able to ameliorate relative RV precision from 300 m/s to 200 m/s. This was a substantial improvement in our ability to detect gas-giant planets. However none of the 15 dwarfs we monitored indicate a presence of companions. This null result was then used to compute the upper limit to the binary frequency and close-in Jupiter mass planetary frequency. We find the binary frequency to be 11% while the planetary frequency was 1.20%.