A jigsaw puzzle is a recreational activity that involves assembling a certain number of pieces into a combined and well-fitting unit without creating gaps between adjacent pieces. Two-dimensional puzzles are divided into two main categories, the “apictorial” in which the only information available is the shape of the pieces and the “pictorial” which may take into account not only the shape of the pieces, but also their content. Jigsaw puzzles are considered as one of the most popular category of puzzles. The majority of them are accompanied by a guiding image and there is only one “counterpart” for each side of each piece (pictorial jigsaw puzzles), although some more difficult variants have blank pieces, the so-called apictorial jigsaw puzzles. In this paper, we will examine the open problem of solving pictorial and apictorial jigsaw puzzles, and their various applications, such as the reconstruction of two-dimensional fragmented objects, the restoration of fragmented wall-paintings and the repair of shredded documents. We will also present an evaluation of the state-of-the- art jigsaw puzzle reassembly techniques in pictorial and apictorial puzzles.