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This paper introduces a new kind of mosaic, called Jigsaw Image Mosaic (JIM), where image tiles of arbitrary shape are used to compose the final picture. The generation of a Jigsaw Image Mosaic is a solution to the following problem: given an arbitrarily-shaped container image and a set of arbitrarily-shaped image tiles, fill the container as compactly as possible with tiles of similar color to the container taken from the input set while optionally deforming them slightly to achieve a more visually-pleasing effect. We approach the problem by defining a mosaic as the tile configuration that minimizes a mosaicing energy function. We introduce a general energy-based framework for mosaicing problems that extends some of the existing algorithms such as Photomosaics and Simulated Decorative Mosaics. We also present a fast algorithm to solve the mosaicing problem at an acceptable computational cost. We demonstrate the use of our method by applying it to a wide range of container images and tiles.