We present the conformal magnifier, a novel interactive focus+context visualization technique that magnifies a region of interest (ROI) using conformal mapping. Our framework supports the arbitrary shape design of magnifiers for the user to enlarge the ROI while globally deforming the context region without any cropping. By using the mathematically well-defined conformal mapping theory and algorithm, the ROI is magnified with local shape preservation (angle distortion minimization), while the transition area between the focus and context regions is deformed smoothly and continuously. After the selection of a specified magnifier shape, our system can automatically magnify the ROI in real time with full resolution even for large volumetric data sets. These properties are important for many visualization applications, especially for the computer aided detection and diagnosis (CAD). Our framework is suitable for diverse applications, including the map visualization, and volumetric visualization. Experimental results demonstrate the effectiveness, robustness, and efficiency of our framework.