The vasculature plays a critical role in homeostasis and health as well as in the development and progression of a wide range of diseases, including cancer, cardiovascular diseases, metabolic diseases (and their complications), chronic inflammatory diseases, ophthalmic diseases, and neurodegenerative diseases. As such, the growth of the vasculature mediates normal development and physiology, as well as disease, when pathologically induced vessels are morphologically and functionally altered owing to an imbalance of angiogenesis-stimulating and angiogenesis-inhibiting factors. This review offers an overview of the angiogenic process and discusses recent findings that provide additional interesting nuances to this process, including the roles of intussusception and angiovasculogenesis, which may hold promise for future therapeutic interventions. In addition, we review the methodology, including those of in vitro and in vivo assays, which has helped build the vast amount of knowledge on angiogenesis available today and identify important remaining knowledge gaps that should be bridged through future research.