Tiling Theory is a branch of mathematics that studies how shapes can cover a surface without gaps or overlaps. This involves using geometric figures, called tiles, which can be regular or irregular. The goal is to understand the arrangements and patterns that can be formed, leading to insights in areas like combinatorics and geometry. One famous example in Tiling Theory is the Penrose tiling, which uses a set of shapes to create non-repeating patterns. Tiling has applications in various fields, including computer graphics, architecture, and even art, where the arrangement of tiles can create visually appealing designs.