Strain is powerful for discovery and manipulation of new phases of matter; however, elastic strains accessible to epitaxial films and bulk crystals are typically limited to small ( < 2 %), uniform, and often discrete values. This Perspective highlights emerging directions for strain and strain gradient engineering in free-standing single-crystalline membranes of quantum materials. Membranes enable large ( ∼ 10 %), continuously tunable strains and strain gradients via bending and rippling. Moreover, strain gradients break inversion symmetry to activate polar distortions, ferroelectricity, chiral spin textures, superconductivity, and topological states. Recent advances in membrane synthesis by remote epitaxy and sacrificial etch layers enable extreme strains in transition metal oxides, intermetallics, and Heusler compounds, expanding beyond the natively van der Waals (vdW) materials like graphene. We highlight emerging opportunities and challenges for strain and strain gradient engineering in membranes of non-vdW materials.