Low nitrogen availability is one of the main limiting factors for plant growth and development, and high doses of N fertilizers are necessary to achieve high yields in agriculture. However, most N is not used by plants and pollutes the environment. This situation can be improved by enhancing the nitrogen use efficiency (NUE) in plants. NUE is a complex trait driven by multiple interactions between genetic and environmental factors, and its improvement requires a fundamental understanding of the key steps in plant N metabolism—uptake, assimilation, and remobilization. This review summarizes two decades of research into bioengineering modification of N metabolism to increase the biomass accumulation and yield in crops. The expression of structural and regulatory genes was most often altered using overexpression strategies, although RNAi and genome editing techniques were also used. Particular attention was paid to woody plants, which have great economic importance, play a crucial role in the ecosystems and have fundamental differences from herbaceous species. The review also considers the issue of unintended effects of transgenic plants with modified N metabolism, e.g., early flowering—a research topic which is currently receiving little attention. The future prospects of improving NUE in crops, essential for the development of sustainable agriculture, using various approaches and in the context of global climate change, are discussed.