The application of the Immersed Boundary Method (IBM) coupled with Adaptive Mesh Refinement (AMR) is considered to be one of the powerful tools for solving complex viscous incompressible flow problems. In this paper, the Kajishima-cut-cell IBM is combined with AMR to solve the flow of complex incompressible, viscous fluid, and rigid body problems. In the IBM, the solid and fluid motions at the interface are controlled by a body force that can be calculated with a fraction of solid volume. The objective is to develop an automatic adaptive mesh refinement strategy to enhance the solution in proximity to the fluid-structure interface. This is necessary as the flow field might be significantly affected by the structure boundaries; therefore, it is essential to capture the boundary layers precisely. The capability of this method can be demonstrated through computational results to improve the flow resolution near the fluid structure. The proposed approach is validated using a 2D laminar flow numerical example. The approach is validated in terms of accuracy and performance. The combined IBM-Adaptive mesh refinement approach showed a promising outcome for the investigated problem. The result of the implemented method achieved an acceptable error performance within a reasonably low computation time. © 2019 Sharif University of Technology. All rights reserved.