Vector database management systems have emerged as an important component in modern data management, driven by the growing importance for the need to computationally describe rich data such as texts, images and video in various domains such as recommender systems, similarity search, and chatbots. These data descriptions are captured as numerical vectors that are computationally inexpensive to store and compare. However, the unique characteristics of vectorized data, including high dimensionality and sparsity, demand specialized solutions for efficient storage, retrieval, and processing. This narrative literature review provides an accessible introduction to the fundamental concepts, use-cases, and current challenges associated with vector database management systems, offering an overview for researchers and practitioners seeking to facilitate effective vector data management.