The use of mmWave band promises to revolutionize the communication technology that will came in 5G and beyond systems. Future communication systems will stretch the current technology beyond its fundamental limits and existing applications, and will enable new society transforming applications in IoT-based devices, M2M communications, self-driving vehicles, V2V, and V2X networks to name some. While mmWave band allows for implementing antenna arrays of a very large aperture and electrical size in relatively small space that enables all the aforementioned applications, it imposes fundamental challenges on required signal processing techniques. The prime goal of the proposed research program is to develop and advance new signal processing techniques and theories suitable and best adopted for mmWave systems. We anticipate that one of the most significant breakthroughs in this path will be the development of signal processing theory and methods for sparse arrays, especially in mmWave band.