Cartograms are a well-known technique for showing geography-related statistical information, such as demographic and epidemiological data. The idea is to distort a map by resizing its regions according to a statistical parameter, but in a way that keeps the map recognizable. In this thesis, we formally define a family of cartogram drawing problems. We show that even simple variants are unsolvable in the general case. Furthermore, we propose two methods of cartogram generation. Both algorithms cover a wide range of user requirements that can be directly maintained by the user. The first algorithm strictly retains the topology and shape of the map regions while minimizing the area error. The second algorithm approximates the map regions by rectangles focusing on an exact area approach and minimizing the shape, topology, and relative position of the map regions. Both algorithms are fast enough to be used in interactive systems which is important to be used as an information visualization technique. Application experiments show that the proposed algorithms can compute high-quality cartograms in a few seconds, even for large maps with a high number of polygons. Also, our algorithms were designed for a dynamic visualization, for which we need an algorithm that recalculates a cartogram in a few seconds. None of the algorithms published before provides adequate performance with an acceptable level of quality for this application. Additional application examples of the two new algorithms introduced in this work help to demonstrate their potential.