In this thesis the structural, elastic and electronic properties as well as electric eld gradient of the strongly correlated intermetallics LnX3 (Ln = La-Gd, X = In, Sn) have been investigated by using the full potential linearized augmented plane wave plus local orbital (FP-LAPW+lo) method within the framework of density functional theory. The structural properties of these compounds are calculated by LDA, GGA, meta-GGA, WC, B3PW91, LDA+U and GGA+U schemes. The calculated lattice parameters are found consistent with the available experimental results. Our results show contraction of lattice constant along the series and the divalent state of Eu is also veri ed. The itinerant and localized behavior of electrons in f -states of these compounds is also discussed. These compounds have 4f orbitals and hence strong electron-electron correlation e ect is expected, therefore, the electronic properties are also calculated with the Hubbard potential U (GGA + U and LDA + U) and the e ect of Hubbard potential on the density of states is discussed in details. The relativistic e ects are also considered by including the spin-orbit coupling (SOC). The spin-orbit coupling predict the correct electronic properties and also reveals the splitting of 4f states of the rare-earth elements. It a ects the band structures of the compounds and induces non-degeneracies in some degenerate states in the vicinity of the Fermi level. Furthermore, the SOC e ect increases from left to right in the lanthanide series in the LnIn3 and LnSn3, which shows interesting nature of SOC e ect in the periodic table. The elastic constants of these compounds are also calculated. Our calculated values for the elastic constants of the compounds are closer with the available experimental values as compared to the other theoretical results. The mechanical properties for the compounds under studies such as shear iii modulus, bulk modulus, Young''s modulus, Kleinman parameters, anisotropic ratio, Poisson''s ratio, Lame''s coe cients are also determined . The Cauchy pressure and B/G ratio are also investigated to evaluated the ductile and brittle nature of LnIn3 and LnSn3 compounds. The Sound velocities for shear and longitudinal waves, and Debye temperature also explored on the basis of mechanical properties. The Electriceld gradients (EFG) are also calculated for the rare-earth intermetallics LnIn3 and LnSn3 (Ln = La, Ce, Pr and Nd) using the GGA, GGA + U, as well as GGA + SOC. Our results show that the EFGs calculated by GGA+U approach are in better agreement with the available experimental values of the M ossbauer spectroscopy as compared to the other theoretical schemes. Our results show nonzero EFGs at rare-earth sites in the AFM phase. The present research will add some theoretical understanding of these materials and will alsoll the gap about some of the physical properties of these compounds in literature.