Climate change, population growth, and increased industrial activities are exacerbating freshwater scarcity and leading to increased interest in desalination of saline water. Brackish water is an attractive alternative to freshwater due to its low salinity and widespread availability in many water-scarce areas. However, partial or total desalination of brackish water is essential to reach the water quality requirements for a variety of applications. Selection of appropriate technology requires knowledge and understanding of the operational principles, capabilities, and limitations of the available desalination processes. Proper combination of feedwater technology improves the energy efficiency of desalination. In this article, we focus on pressure-driven and electro-driven membrane desalination processes. We review the principles, as well as challenges and recent improvements for reverse osmosis (RO), nanofiltration (NF), electrodialysis (ED), and membrane capacitive deionization (MCDI). RO is the dominant membrane process for large-scale desalination of brackish water with higher salinity, while ED and MCDI are energy-efficient for lower salinity ranges. Selective removal of multivalent components makes NF an excellent option for water softening. Brackish water desalination with membrane processes faces a series of challenges. Membrane fouling and scaling are the common issues associated with these processes, resulting in a reduction in their water recovery and energy efficiency. To overcome such adverse effects, many efforts have been dedicated toward development of pre-treatment steps, surface modification of membranes, use of anti-scalant, and modification of operational conditions. However, the effectiveness of these approaches depends on the fouling propensity of the feed water. In addition to the fouling and scaling, each process may face other challenges depending on their state of development and maturity. This review provides recent advances in the material, architecture, and operation of these processes that can assist in the selection and design of technologies for particular applications. The active research directions to improve the performance of these processes are also identified. The review shows that technologies that are tunable and particularly efficient for partial desalination such as ED and MCDI are increasingly competitive with traditional RO processes. Development of cost-effective ion exchange membranes with high chemical and mechanical stability can further improve the economy of desalination with electro-membrane processes and advance their future applications.