Shape-changing arrays are an emerging frontier of phased array development. These arrays fold, conform, and flex dynamically as they operate. In this work we describe the technology developments which have enabled their creation and use. We present the theoretical implications of aperture change for arrays and methods for taking advantage of these aperture changes. We discuss existing shape-changing array components and systems. Operation techniques for shape-changing arrays, including new results demonstrating a method for determining the shape of an asymmetrically bent flexible array using only the mutual coupling between elements, are shown. Finally, we present a comparison of shape-changing systems across a variety of physical and electrical metrics.