Quantum memory, which maps photonic quantum information into a stationary medium and retrieves it at a chosen time, plays a vital role in the advancement of quantum information science. In particular, the scalability of a quantum memory is a central challenge for quantum network that can be overcome by using integrated devices. Quantum memory with an integrated device is highly appealing since it not only expands the number of memories to increase data rates, but also offers seamless compatibility with other on-chip devices and existing fiber network, enabling scalable and convenient applications. Over the past few decades, substantial efforts have been dedicated to achieving integrated quantum memory using rare earth ions doped solid-state materials, color centers, and atomic gases. These physical platforms are the primary candidates for such devices, where remarkable advantages have been demonstrated in achieving high-performance integrated quantum memory, paving the way for efficiently establishing robust and scalable quantum network with integrated quantum devices. In this paper, we aim to provide a comprehensive review of integrated quantum memory, encompassing its background and significance, advancement with bulky memory system, fabrication of integrated device, and its memory function considering various performance metrics. Additionally, we will address the challenges associated with integrated quantum memory and explore its potential applications. By analyzing the current state of the field, this review will make a valuable contribution by offering illustrative examples and providing helpful guidance for future achievements in practical integrated quantum memory.