Shift operators are important operators in functional analysis in the study of other operators like compact operators, finite operators, normal opera-tors among others. Left shifts, right shifts and unilateral shifts have been studied in details with useful results obtained, However, little work has been done with regard to their adjoints and spectra in infinite dimen¬sional complex Hilbert spaces. A comparison of the spectra of bilateral shifts with those of compact operators have not been considered. The objectives of this study therefore were_ to: Investigate the spectrum of bi¬lateral shifts: Investigate the spectrum of compact operators: Investigate the relationship between the spectrum of bilateral shifts and spectrum of compact operators. Methodology involved stating known fundamen¬tal results without their proofs. We also used the technical approaches of tensor products, direct sum and polar decomposition in investigating the spectral properties. Certain inequalities were also useful for example triangle inequality, Holder's inequality, Cauchy-Schwartz among others. The results in this study show that zero is contained in the point spec¬trum of operator M if and only if M is a bounded one to one self adjoint compact operator, the spectrum is non empty if and only if operator T is of finite rank, if the point spectrum of an operator is nonempty then its approximate point spectrum is connected. The results also show that the spectrum of a bilateral shift operator and a compact operator on a real space are equal. The study of bilateral shifts and compact operators is useful in the areas of electrical engineering, transport sector, quantum computing and in medical imaging.