The adhesive contact problem of a half-space indented by a rigid cylindrical punch in the theory of couple-stress elasticity is investigated. Both the size effect and adhesion are considered to analyze the two-dimensional frictionless contact problem. The method of Mindlin's stress function and the technology of Fourier transform are applied to reduce the mixed boundary value problem to the singular integral equations. Efficient computational techniques is used to obtain the influence of the size effect and adhesion parameter on the adhesive contact behavior. Results indicated that the size effect play an important role in contact with adhesion and can help understanding the mechanism of adhesion.