An old dilemma in structural shape optimization is the needed tight link between design model or geometric description and analysis model. The intention of this paper is to show that isogeometric analysis offers a potential and promising way out of this dilemma. To this end we show a structural shape optimization framework based on the isogeometric analysis approach. With the discretization based on Non-Uniform Rational B-Splines (NURBS) the analysis model represents the structural geometry exactly. Furthermore, NURBS enable efficient geometry control together with smooth boundaries. They are the de facto standard in CAD systems, but are also widely used in a shape optimal design context to define the geometry representation and the design variables. With the presented isogeometric approach to shape optimization, the analysis model is inherently merged with the design model, omitting the typically involved interplay between both. We derive analytical sensitivities for NURBS discretizations which allow application of efficient gradient-based optimization algorithms. The present contribution is restricted to two-dimensional problems of linear elasticity, but the extension to three dimensions and other problem classes is straightforward. Some representative examples demonstrate and validate the methodology. Further, the potential of boundary continuity control within isogeometric structural shape optimization is explored to trigger smooth or less smooth (angular) designs.