Digital data, generated by corporate and individual users, is growing day by day due to a vast range of digital applications. Magnetic hard disk drives (HDDs) currently fulfill the demand for storage space, required by this data growth. Although flash memory devices are replacing HDDs in applications like mobile phones, laptops, and desktops, HDDs cover the majority of digital data stored in the cloud and servers. Since the capacity growth of HDDs is slowing down, it is essential to look for a potential alternative. One such alternative is domain wall (DW) memory, where magnetic domains in the form of two-dimensional or three-dimensional wires are used to store the information. DW memory (DWM) devices should satisfy the four basic operations, such as writing (nucleating domains or inserting DWs in memory element), storing (stabilizing DWs), shifting (moving DWs), and reading (reading magnetization direction). An external magnetic field or spin-transfer torque can be used to write the information. Spin–orbit torque or electric field may be used for shifting the DWs. The information can be read using tunneling magnetoresistance. The domains may be stored along the tracks using artificial pinning potentials. The absence of moving parts makes the DWM consume less power as compared to HDDs, and be more robust. The potential to stack many layers to store information in three dimensions makes them potentially a large storage capacity device. In addition to memory, DW devices also offer a route for making synaptic devices for neuromorphic computing. Despite these potential advantages of DWM, significant advances in research are needed before DWM could become commercially viable. One of the major challenges associated with DWM is DW dynamics. Many problems, such as controlled DW motion, the stability of domains, reducing the dimensions of the DW devices are still to be addressed. Artificial pinning sites fabricated through either geometrical or non-geometrical methods have been proposed for controlling DW motion. This review paper presents a survey of the investigations carried out so far and the future perspective of such devices.