DESIRE: Diagonally extended singly implicit Runge-Kutta effective order methods. In the last approximatly 20 years many classes of singly-implicit Runge-Kutta methods (SIRK) have been developed. More general classes are the so-called “diagonally extended singly implicit” (DESI) methods. A generalization in a different direction are the effective SIRK methods (ESIRK). The new generalization, which is the subject of this paper, includes all the above-mentioned classes and is denoted by the acronym DESIRE (diagonally extended singly-implicit Runge-Kutta effective order methods). The idea is to combine the use of additional diagonally-implicit stages with a relaxation of the order conditions in the effective order.par Even though a full implementation of DESIRE methods is not yet available, the authors believe that they are very promising alternatives to standard methods for solving stiff equations. Some preliminary numerical results for the new methods are presented in the article.