The full theory of the finite-dimensional representations of the Lie algebra of the Lorentz group is deduced using the general framework of the representation theory of semisimple Lie algebras. The finite-dimensional representations of the connected component SO ( 3 ; 1 ) + of the full Lorentz group O(3; 1) are obtained by employing the Lie correspondence and the matrix exponential. The full finite-dimensional representation theory of the universal covering group (and also the spin group, a double cover) SL ( 2 , C ) of SO ( 3 ; 1 ) + is obtained, and explicitly given in terms of action on a function space in representations of SL ( 2 , C ) and s l ( 2 , C ) . The representatives of time reversal and space inversion are given in space inversion and time reversal, completing the finite-dimensional theory for the full Lorentz group. The general properties of the (m, n) representations are outlined. Action on function spaces is considered, with the action on spherical harmonics and the Riemann P-functions appearing as examples. The infinite-dimensional case of irreducible unitary representations are realized for the SL ( 2 , C ) principal series and the complementary series. Finally, the Plancherel formula for SL ( 2 , C ) is given, and representations of SO(3, 1) are classified and realized for Lie algebras.