Many traditional quilt designs are based on tilings by squares. Most of these traditional patterns use edge-to-edge tilings, meaning that all of the corners and edges of each tile are aligned with all of the vertices and edges of the tiling. Using non-edge-to-edge tilings as the basis of a quilt design allows for many new quilt settings that are not typically found in traditional quilts, while maintaining the convenience and tradition of using square quilt blocks. This paper discusses the mathematics of non-edge-to-edge tilings by squares, the design of such tilings using Microsoft Excel, and the technical details behind physical construction of such quilts. Both infinite tilings of the plane and finite tilings of rectangles are discussed.