Two-dimensional conjugated metal-organic frameworks (2D c-MOFs) have emerged as a new class of crystalline layered conducting materials that hold significant promise for applications in electronics and spintronics. However, current 2D c-MOFs are mainly made from organic planar ligands, whereas layered 2D c-MOFs constructed by curved or twisted ligands featuring novel orbital structures and electronic states remain less developed. Herein, we report a Cu-catecholate wavy 2D c-MOF (Cu3(HFcHBC)2) based on a fluorinated core-twisted contorted hexahydroxy-hexa-cata-hexabenzocoronene (HFcHBC) ligand. We show that the resulting film is composed of rod-like single crystals with lengths up to ∼4 μm. The crystal structure is resolved by high-resolution transmission electron microscopy (HRTEM) and continuous rotation electron diffraction (cRED), indicating a wavy honeycomb lattice with AA-eclipsed stacking. Cu3(HFcHBC)2 is predicted to be metallic based on theoretical calculation, while the crystalline film sample with numerous grain boundaries apparently exhibits semiconducting behavior at the macroscopic scale, characterized by obvious thermally activated conductivity. Temperature-dependent electrical conductivity measurements on the isolated single-crystal devices indeed demonstrate the metallic nature of Cu3(HFcHBC)2, with a very weak thermally activated transport behavior and a room-temperature conductivity of 5.2 S cm-1. Furthermore, the 2D c-MOFs can be utilized as potential electrode materials for energy storage, which display decent capacity (163.3 F g-1) and excellent cyclability in an aqueous 5 M LiCl electrolyte. Our work demonstrates that wavy 2D c-MOF using contorted ligands are capable of intrinsic metallic transport, marking the emergence of new conductive MOFs for electronic and energy applications.