In this thesis, we present theoretical studies of antiperovskites ANCa3 (A=Ge, Sn, Pb), BCFe3 (B=Al, Zn, Ga), SnCD3 (D=Co and Fe) and MXY3 (M=Al, Ga, Ir, Mg, Pd, Pt, Rh; X=C, N; Y=Mn, Ni, Sc, Ti, Cr, Fe) as well as SbNCa3, BiNCa3, SbNSr3 and BiNSr3. The calculations are carried out with the full-potential linearized augmented plane waves plus local orbital (FPLAPW+lo) method within the framework of density functional theory (DFT) as well as Boltzmann’s theory. The exchange–correlation effects are treated by the local density approximation (LDA), generalized gradient approximation (GGA-PBEsol) and Engel and Vosko GGA (EV-GGA). Furthermore, the modified Becke and Johnson (mBJ) as well as improved mBJ potentials are used for the exact band gaps of the semiconductors. The relativistic effects in some of the compounds under study are explored by spin-orbit coupling. The consistency of the calculated results of the thermoelectric properties of SnCCo3 and SnCFe3 with the experimental results confirms the reliability of our theoretical calculations for the other investigated metallic antiperovskites, ANCa3 (A=Ge, Sn, Pb), BCFe3 (B=Al, Zn, Ga), SnCD3 (D=Co and Fe) and MXY3 (M=Al, Ga, Ir, Mg, Pd, Pt, Rh; X=C, N; Y=Mn, Ni, Sc, Ti, Cr, Fe). Our results for ANCa3 (A=Ge, Sn, Pb), BCFe3 (B=Al, Zn, Ga) and SnCD3 (D=Co, Fe) indicate that the thermopower of these materials can be enhanced by changing the chemical potential. The dimensionless figure of merit for the three nitrides approaches to 0.96 at room temperature, which predicts the usefulness of these materials in thermoelectric devices. Furthermore, the thermal conductivity of these compounds is minimum at room temperature for chemical potential of -0.25 eV to 0.25 eV, with maximum values of dimensionless figure of merit in this range. The striking feature of these studies is identifying a metallic compound, SnNCa3, with the highest value of Seebeck coefficient at room temperature out of all metals. Furthermore, electronic and thermoelectric properties of carbon and nitrogen based twenty metallic antiperovskites MXY3 (M=Al, Ga, Ir, Mg, Pd, Pt, Rh; X=C, N; Y=Mn, Ni, Sc, Ti, Cr, Fe) are investigated. We find high values of Seebeck coefficient and small values of electronic thermal conductivity for AlCTi3, AlNSc3, AlCNi3, AlNTi3, GaCCr3 and MgCNi3 between -0.25 and 0.25 eV chemical potential. These results show high dimensionless figure of merit in metallic materials and therefore, we predict these materials can be potential candidates for low temperature thermoelectric applications. Figure of merit for AlNTi3, GaCCr3, AlCNi3, AlNSc3, MgCNi3 and AlCTi3 materials reaches to 0.32, 0.25, 0.19, 0.19, 0.2 and 0.25 respectively, and hence are predicted to be low temperature thermoelectric materials. The structural, electronic and optical properties of antiperovskite semiconductors, SbNCa3, BiNCa3, SbNSr3 and BiNSr3 are also studied. The calculated lattice constants for these compounds are found consistent with the available experimentally measured values and other theoretical results. The band profiles show that all of these materials are direct band gap semiconductors with the band gap values of 1.1 eV, 1.09 eV, 0.92 eV and 0.81 eV for SbNCa3, BiNCa3, SbNSr3 and BiNSr3 respectively. The direct band gap nature reveals that they may be effective in optical devices and therefore the optical properties of these compounds like the real and imaginary parts of dielectric function, refractive index and absorption coefficient are calculated and discussed. Furthermore, the thermoelectric properties of these semiconductors are also calculated. Our results show high values of Seebeck coefficient for these materials between -0.25 eV and 0.25 eV chemical potential values.