This paper presents a systematic generalization of the quantum geometric condition for the lowest Landau level (LL) to arbitrary higher LLs. The authors introduce a set of single-particle states called "generalized Landau levels" (GLLs), which exhibit exactly quantized values of the integrated trace of quantum metric, regardless of the non-uniformity of their quantum geometric quantities. These GLLs maintain the same quantized integrated trace value as standard LLs, determined by their corresponding LL indices. The quantum geometries of individual and multiple GLLs are explicitly derived, and the results are understood in terms of holomorphic curves and moving frames. The GLLs form a complete basis for generic topological bands. The authors propose a model constructed from GLLs that resembles the key features of the standard first LL. Using this model, they identify a region of single-particle geometric quantities that permit the Moore-Read non-Abelian fractionalized phase. The authors also show the existence of a first GLL type narrow band and a zero-field Moore-Read state at the second magic angle in a double twisted bilayer graphene model. They conclude that GLLs will serve as a systematic tool for analyzing topological Chern bands and fractionalized phases.This paper presents a systematic generalization of the quantum geometric condition for the lowest Landau level (LL) to arbitrary higher LLs. The authors introduce a set of single-particle states called "generalized Landau levels" (GLLs), which exhibit exactly quantized values of the integrated trace of quantum metric, regardless of the non-uniformity of their quantum geometric quantities. These GLLs maintain the same quantized integrated trace value as standard LLs, determined by their corresponding LL indices. The quantum geometries of individual and multiple GLLs are explicitly derived, and the results are understood in terms of holomorphic curves and moving frames. The GLLs form a complete basis for generic topological bands. The authors propose a model constructed from GLLs that resembles the key features of the standard first LL. Using this model, they identify a region of single-particle geometric quantities that permit the Moore-Read non-Abelian fractionalized phase. The authors also show the existence of a first GLL type narrow band and a zero-field Moore-Read state at the second magic angle in a double twisted bilayer graphene model. They conclude that GLLs will serve as a systematic tool for analyzing topological Chern bands and fractionalized phases.