Acentric materials with bulk photovoltaic effect (BPVE) have recently emerged as a class of promising optoelectronic materials due to broken inversion symmetry. In this review, we mainly aim to provide a comprehensive understanding of acentric single crystals for BPVE-based optoelectronic applications. An overview of research background and recent progress of the acentric materials with BPVE is described. Moreover, the characteristics of BPVE and BPVE-based optoelectronic device applications are illustrated in detail. Finally, possible future research directions and challenges of the acentric single crystals and their optoelectronic devices with BPVE characteristics are proposed with our viewpoints in the emerging optoelectronic field.