Is there any reason why we should predict contact maps (CMs)? The question is one of the several 'NP-hard' questions that arise when striving for feasible solutions of the protein folding problem. At some point, theoreticians started thinking that a possible alternative to an unsolvable problem was to predict a simplified version of the protein structure: a CM. In this chapter, we will clarify that whenever problems are difficult they remain at least as difficult in the process of finding approximate solutions or heuristic approaches. However, humans rarely give up, as it is stimulating to find solutions in the face of difficulties. CMs of proteins are an interesting and useful representation of protein structures. These two-dimensional representations capture all the important features of a protein fold. We will review the general characteristics of CMs and the methods developed to study and predict them, and we will highlight some new ideas on how to improve CM predictions.