Computer pioneers have correctly predicted that programmers would want unlimited amounts of memory. An economical solution to this desire is the implementation of a Memory Hierarchical System, which takes advantage of locality and cost/performance of memory technologies. As time has gone by, the technology has progressed, bringing about various changes in the way memory systems are built. Memory systems must be flexible enough to accommodate various levels of memory hierarchies, and must be able to emulate an environment with unlimited amount of memory. For more than two decades the main emphasis of memory system designers has been achieving high performance. However, recent market trends and application requirements suggest that other design goals such as low-power, predictability, and flexibility/ reconfigurability are becoming equally important to consider. This paper gives a comprehensive overview of memory systems with the objective to give any reader a broad overview. Emphasis is put on the various components of a typical memory system of present-day systems and emerging new memory system architecture trends. We focus on emerging memory technologies, system architectures, compiler technology, which are likely to shape the computer industry in the future.