1. Schematic representation of transport mechanisms in membranes, where D_c is the location of carbon centers, and the pore diameter is defined as D_p = D_c - D_vdw/$\sqrt{2}$, where D_vdw is the van der Waals diameter of a carbon atom^{[1,10,11,14-17]}

2. Schematic of pores in graphene sheet (a-b) and pore electron density isosurfaces of porous graphene (c-d)^[28]

3. Snapshots of passing-through events of gas molecules^[29]

4. Schematic of fabrication of a large-area graphene membrane by the nanoporous carbon film-assisted transfer method (a) and high-resolution TEM images of the intrinsic defects in graphene lattice (b)^[33]

5. Photo of a 1-µm-thick GO film (a), electron micrograph of the film’s cross section (b), schematic view for possible permeation through the laminates (c), and examples of He-leak measurements for a freestanding submicrometer-thick GO membrane and a reference PET film (d)^[40] (1 mbar=100 Pa)

6. Schematic illustration of two different GO coating methods (Method one and Method two) (a), schematic illustration of possible gas transport through graphene membrane (b), AFM images of GO membrane surfaces with insert images showing depth profiles of GO membrane surfaces: (c) method one [root mean square roughness (R_q) = 0.8 nm, average roughness (R_a) = 0.614 nm] and (d) method two (R_q = 0.608 nm, R_a = 0.467 nm)^[19]

7. Schematic representation of the assembly of GO nanosheets in polymeric environment (a), digital photographs of the membrane with 0.1wt% GO (b), overview (the yellow dashed lines are eye-guiding lines indicating the GO laminates in these regions) (c) and expanded TEM image of the cross section of GO-1 membrane (d)^[17]

8. Force analysis for one 2D channel unit consisting of GO nanosheets and polymer chain (a), schematic illustration of intrinsic force induced disordered structure (left) and highly ordered laminar structures (right) driven by introduced synergistic external forces (b)^[54]

9. Schematic illustration of the gas solution-diffusion transport pathway through a GO-SILM (a), molecular structures of [BMIM][BF₄] IL confined in the GO nanochannels (b), the cross section of the GO membrane (c) and GO-SILM (d), respectively^[55]

10. Schematic of gas permeation pathway across MoS₂ membranes before and after heating at 160 ℃(a)^[63], cross section SEM images of the membrane (b-e)^[64], and synthesis process for MoS₂-SILM (f)^[69](b) GO membrane; (c) MoS₂ membrane; (d) GO-MoS₂ (50/50) hybrid membrane; (e) GO-MoS₂ (75/25) hybrid membrane

11. SEM image of the delaminated MXene nanosheets on porous anodic aluminum oxide (AAO) (a) with insert showing the Tyndall scattering effect in MXene colloidal solution in water, cross-sectional SEM image of the MXene membrane (b) with insert showing a tweezer bent membrane, AFM image of the Mxene nanosheet on cleaved mica (c), illustration of the spacing between the neighboring MXene nanosheets in the membrane (d)^[77], structures and gas transport of H₂-selective and CO₂-selective Mxene nanofilms (e)^[78]