Verifiable computation (VC) paradigm has got the captivation that in real term is highlighted by the concept of third party computation. In more explicate terms, VC allows resource constrained clients/organizations to securely outsource expensive computations to untrusted service providers, while acquiring the publicly or privately verifiable results. Many mainstream solutions have been proposed to address the diverse problems within the VC domain. Some of them imposed assumptions over performed computations, while the others took advantage of interactivity/non-interactivity, zero knowledge proofs, and arguments. Further proposals utilized the powers of probabilistic checkable or computationally sound proofs. In this survey, we present a chronological study and classify the VC proposals based on their adopted domains. First, we provide a broader overview of the theoretical advancements while critically analyzing them. Subsequently, we present a comprehensive view of their utilization in the state of the art VC approaches. Moreover, a brief overviewof recent proof based VC systems is also presented that lifted up the VC domain to the verge of practicality. We use the presented study and reviewed results to identify the similarities and alterations, modifications, and hybridization of different approaches, while comparing their advantages and reporting their overheads. Finally, we discuss implementation of such VC based systems, their applications, and the likely future directions.