In our research, we study IT-systems for highly skilled professionals in complex and dynamic work situations. Such situations can be found in e.g. health care, process and traffic control and in administration. The demands on the operators/users are often very high concerning quality performance, efficiency, timeliness, safety, communication and cooperation. Our experience shows that human operators can overview, interpret and in real time use an almost unlimited amount of information, if it is relevant to the situation and visualized according to human capabilities. The solution to the visualization problem is therefore not to avoid or hide complexity, but to cope with it, to accept that the complexity must be there. The challenge is to develop systems for visualization and support, which can be used efficiently in relation to the complexity of the work task. We believe in recognizing complexity. First, we describe the scientific foundation of such an approach. Second, we give a detailed example of a complex visualization problem, emphasizing the demanding cognitive operations the operators have to conduct. Finally, we describe the solutions, the visualizations and interactions that make it possible to support the cognitively demanding task, taking care of the complexity without losing the rich amount of information necessary for the operators in different situations, but without adding unnecessary complexity in terms of complicated handling of the user interface and the information systems. Some of these visualizations now run in real systems and have been evaluated, and we end up by suggesting recommendations for successful visualizations in complex work tasks.