Inverse design methods are powerful tools for aerodynamic shape design. Inverse design includes determining a shape of body so that a prescribed pressure, velocity or shear stress distribution is achieved on its surface. In this thesis, a new inverse design algorithm called Elastic Surface Algorithm (ESA) is developed and enhanced for axial-flow compressor blade design in subsonic and transonic flow regimes with separation. ESA is an iterative inverse design method which includes two parts: an efficient inverse design algorithm, called elastic surface algorithm that uses a 2-D nonlinear finite element model, and a 2-D flow solver. In this thesis, the Fluent commercial software is used as a compressible viscous flow solver. In the proposed method, the airfoil wall shape is changed under a physical algorithm based on the deformation of a flexible beam. The difference between target and current pressure distributions and the difference between target and current wall shear stress distributions cause to deform the flexible beam at each shape modification step and then the internal stresses are set to zero. When current pressure and wall shear stress distributions converge to the target ones, the deformation of airfoil walls is stopped and the final airfoil shape is achieved. In this investigation, the rotor and stator blades of the first stage of an axial-flow compressor in subsonic and transonic flow regimes are validated by the enhanced ESA, which shows the robustness of the method in flow regime with separation and normal shock. Also, some algorithms are developed to improve the performance of this method for compressor blade designs. Moreover, some design examples of axial-flow compressor blades in subsonic and transonic viscous regimes are presented here, which show the flexibility of the proposed method. Results show that the enhanced Elastic Surface Algorithm is a powerful tool for considering the transonic flow around airfoils. The other advantages of the presented method is its physical approach, high convergence rate and the high ability to utilize computer codes as a black box for flow analysis. Keywords: Inverse Design, Airfoil, Axial-Flow Compressor Blade, Elastic Surface, Viscous Flow Regime, Wall Shear Stress