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Extensive theoretical studies on the ground and excited state properties of systems containing heavy atoms have shown that accurate prediction of transition energies and related properties requires the incorporation of both relativistic and higher order correlation and relaxation effects as these effects are strongly inter- wined. The relativistic and dynamical electron correlation effects can be incor- porated in many-electron systems through a variety of many-body methods like configuration interaction (CI), coupled cluster method (CCM) etc. which are very powerful and effective tool for high precision description of electron correlation in many-electron systems. In this thesis, we investigate the relativistic and correlation effects in heavy atomic and molecular systems using these two highly correlated many-body methods. It is well recognized that, heavy polar diatomic molecules such as BaF, YbF, TlF, PbO, etc. are the leading experimental candidates for the search of violation of Parity (P ) and Time-reversal (T ) symmetry. The experimental detection of such P,T-odd effects in atoms and molecules has important consequences for the theory of fundamental interactions or for physics beyond the standard model (SM). For instance, a series of experiments on TlF have already been reported which provide the tightest limit available on the tensor coupling constant C , proton electric dipole moment (EDM) dp , etc. Experiments on YbF and BaF molecules are also of fundamental significance to the study of symmetry violation in nature, as these experiments have the potential to detect effects due to the electron EDMde. It is therefore imperative that high precession calculations are necessary to interpret these ongoing (and perhaps forthcoming) experimental outcome. For example, the knowledge of the effective electric field E(characterized by Wd) at the unpaired electron is required to link the experimentally determined P,T-odd frequency shift with the electron EDM de. We begin with a brief review of P,T-odd effects in heavy atoms and polar diatomics and the possible mechanisms which can give rise to such effects, in particular, the one arises due to the intrinsic electron EDM de. The P,T-odd interaction constant Wd is computed for the ground (2∑ ) state of YbF and BaF molecules using all-electron DF orbitals at the restricted active space (RAS) CI level. The RASCI space used for both systems in this calculation is sufficiently large to incorporate important core-core, core-valence, and valence-valence electron correlation effects. In addition to Wd, we also report the dipole moment (µe ) for these systems to assess the reliability of the method. The basis set dependency of Wd is also analyzed. The single reference coupled cluster (SRCC) method, developed by the cluster expansion of a single determinant reference function, is one of the most sophisticated method for treating dynamical correlation effects in a size-extensive manner. The non-uniqueness of the exponential nature of the wave operator diversifies the methods in multi-reference context. The multi-reference coupled cluster (MRCC) strategies fall within two broad classes: (a) State-Universal (SU), a Hilbert-space approach and (b) Valence-Universal (VU), a Fock-space approach. In this thesis, we shall be mainly concerned with the VU-MRCC which unlike SU-MRCC uses a single wave operator that not only correlates the reference functions, but also all the lower valence (or the so called subdued) sectors, obtained by deleting the occupancies systematically. The linear response theory (LRT) or equation of motion (EOM) method is another possible alternative which is nowadays extensively used to compute the atomic and molecular properties. Although, the CCLRT or EOM-CC method is not fully extensive in nature, this method has some distinct advantages over the traditional VU-MRCC theory. Further, for one-valence problem like ionization processes, the CCLRT/EOM-CC is formally equivalent to VU-MRCC, and hence, size-extensive. In this thesis, the core-extensive CCLRT and core-valence extensive (all electron) VU-MRCC methods are applied to compute the ground and excited state properties of various atomic and molecular systems (HCl, CuH, Ag, Sr, Yb and Hg) using nonrelativistic and relativistic (for heavy atoms) spinors. The similarities and differences in the structure of these two formalisms are also addressed. We also investigate the ground and excited state properties of HCN which is a system of astrophysical importance. This system has raised interest among the astrophysicists due to its detection in the atmosphere of Titan and Carbon stars. HCN has also been identified via radio-techniques in both comets and interstellar atmosphere. In the ash-photolysis of oxazole, iso-oxazole, and thiozole a transient band system was observed in the region 2500-3050 Å. This band system was attributed to a meta-stable form of HCN, i.e, either HNC or triplet HCN. We carry out detailed theoretical investigations using CCLRT and complete active space self-consistent field (CASSCF) method to characterize this unidentified band and other experimentally observed transitions.