Exfoliation and cleavage create two-dimensional (2D) materials and surfaces with physical and chemical properties distinct from their bulk parents. The rising class of non-van der Waals (non-vdW) 2D materials derived from non-layered crystals provides a fascinating platform, expanding the landscape of low-dimensional materials. Current computational models, however, provide limited guidance: existing descriptors are largely tailored to vdW layered systems. Here, we introduce a general framework predicting crystal cleavage and exfoliable 2D subunits directly from bulk structures. At its core is a universal eXfoliation and Cleavage Potential (XCP) enabling large-scale screening of diverse materials at negligible computational cost. Applying this approach, we obtain 44,030 cleavable surfaces and candidate non-vdW 2D materials from which we investigate – according to our criteria – 2531 likely exfoliable ones using high-throughput density functional theory. A large fraction of these candidates is found to be dynamically and thermodynamically stable, while showing negligible overlap with existing 2D materials databases. Our study thus opens a systematic route to explore and design 2D materials with high chemical and structural diversity.