Physicists and mathematicians have been familiar with differential systems. The fundamental equations used in the modelling of natural phenomena are cast in differential form based on the underlying assumption that spacetime is continuous. While numerical algorithms are often considered as inaccurate approximations of a continuous ‘reality’, there exist domains where discrete systems arise naturally. This chapter deals with a presentation of the various approaches proposed for the detection of integrability, and presents results which will illustrate the parallel existing between integrable discrete and continuous systems. It provides results on systems which either lie between the discrete and continuous ones or go beyond the discrete systems. The chapter discusses a plethora of examples which will serve as a guide for the reader to develop his/her own understanding of the subject. It focuses on two topics which have been among the main themes of the work: integrability detectors and discrete Painleve equations.