In the report an introduction to MIMO systems and Space-Time coding is given, also a motivation for the use of MIMO systems in acoustic underwater communication is given. MIMO system is a way to increase the data rate and security in a transmitted message by using several transmitters and receivers in parallel (Multiple Input Multiple Output). In the report, the theoretical basis behind the transmission capacity in MIMO system is discussed. Methods to achieve maximal capacity for a given channel are given. The background to Space-time coding is given. An example of Space-time codes giving full diversity gain is Alamouti´s coding method. One method to design trellis codes, giving both coding and diversity gain is presented. To simplify the reader´s understanding the mathematical treatment is relatively thorough and given in detail.