Computing has been widely used for data storage and computational purposes. When we discuss about the cloud storage services, the data must be outsourced, so, there may be serious concerns about the authorization and trust management for the cloud service provider (CSP). These concerns are about confidentiality, integrity and access control. In  this paper we are going to discuss various models in brief such as Provable data possession (PDP), Proof of retrievability (POR), HAIL, Attribute Based Encryption Scheme, Plutus, SiRiUS, Third party auditor (TPA) etc that are introduced for addressing such issues about cloud storage systems. Also the mutual trust model given by Ayad Barsoum et al. This scheme supports dynamic data and trust in the cloud computing storage systems.