Non-Euclidean geometries and metrics have aided in the development of theoretical understandings of nontrivial three-dimensional (3D) structures, which are experimentally driven by mechanical instabilities of soft elastic materials. In recent years, two-dimensional (2D) materials have also been shown to accommodate various imaginable mechanical deformations owing to their atomic thickness. In contrast to most previous examples relying on post-growth deformations, we show non-flat configurations of atomically-thin MX2 (M = transition metal and X = chalcogen) via a direct and conformal growth onto pre-defined corrugated surfaces with nonzero Gaussian curvature. Lithographically defined corrugations are tunable in size (50 nm - 1 μm) and curvature, and furthermore, are patternable, hence enabling us to construct laterally connected flat and corrugated networks. We also demonstrate that the grown 2D materials retain their original non-Euclidean architecture even after being suspended. We further explore how 2D crystal growth competes with elastic energy on curved templates employing theoretical calculations and structural analysis.