This thesis concerns the inverse convection problem. The theoretical motivation of this mathematical problem is that it is a generalization of the inverse Calderon conductivity problem. The physical motivation for the study of this mathematical problem is limited to the electrical impedance tomography. In the literature many important contributions concerning the uniqueness, reconstruction and error estimates of the inverse conductivity problem are discussed briefly. This thesis surveys the available methods and results. The emphasis is put on the reconstruction of the inverse convection problem proposed by Tamasan on the following three aspects, where also new results are obtained: Technical Lemmas and estimates of the Forward Problem (Chapter 2): focusing on a new Alessendrini type identity constructed for the inverse convection problem that helps in attaining some technical estimates. These technical estimates leads to some error estimates in the following chapters. The First Regularization (Chapter 3): setting the frame space of work needed to perform regularization. An intermediate regularization is performed using truncation with respect to the first variable. Then, the consequences of this regularization is discussed. Moreover, an error estimate is obtained from this regularization. The Second Regularization (Chapter 4): several steps are performed to complete the regularization method . In addition, we prove the hypothesized method of regularization works well in approximating the convection coefficients from the Dirichlet to Neumann map. Error Estimates (Chapter 5): summary of the regularized reconstruction strategy. Furthermore, error estimates are proved in detailed work and using many lemmas proposed before.