The lateral organ boundaries domain (LBD) genes, as the plant-specific transcription factor family, play a crucial role in controlling plant architecture and stress tolerance. However, the functions of AhLBD genes in the peanut plant (Arachis hypogea L.) remain unclear. In this study, 73 AhLBDs were identified in the peanut plant and divided into three groups by phylogenetic tree analysis. Gene structure and conserved protein motif analysis supported the evolutionary conservation of AhLBDs. Tandem and segment duplications contributed to the expansion of AhLBDs. The evolutionary relationship analysis of LBD gene family between A. hypogaea and four other species indicated that the peanut plant had a close relationship with the soybean plant. AhLBDs played a very important role in response to growth and development as well as abiotic stress. Furthermore, gene expression profiling and real-time quantitative qRT-PCR analysis showed that AhLBD16, AhLBD33, AhLBD67, and AhLBD72 were candidate genes for salt stress, while AhLBD24, AhLBD33, AhLBD35, AhLBD52, AhLBD67, and AhLBD71 were candidate genes for drought stress. Our subcellular localization experiment revealed that AhLBD24, AhLBD33, AhLBD67, and AhLBD71 were located in the nucleus. Heterologous overexpression of AhLBD33 and AhLBD67 in Arabidopsis significantly enhanced tolerance to salt stress. Our results provide a theoretical basis and candidate genes for studying the molecular mechanism for abiotic stress in the peanut plant.