Automatic Repeat-reQuest (ARQ) schemes are extensively used in communication systems and computer networks to achieve reliable transmission. Using space-time codes (STCs) with multiple input multiple output (MIMO) or multiple input single output (MISO) systems is an effective way to combat multipath fading, which is the most severe impairment for wireless communication systems. STCs are designed to use the rich scattering multipath environment provided by using multiple transmit antennas. The work done in this thesis focuses on the use of ARQ schemes with non-orthogonal space-time block codes (NOSTBCs) based on Reed Solomon codes. The truncated-selective ARQ (TS-ARQ) scheme is considered and three novel hybrid ARQ (HARQ) schemes are proposed. Simulation results reveal that, compared to using TS-ARQ with orthogonal space-time block codes (OSTBCs), using NOSTBCs with any of the three proposed HARQ schemes can provide significant gains in terms of dropped packet rate and spectral efficiency at the cost of increased decoding complexity. The performance can be further improved by using the water filling principle to adaptively allocate transmit power among transmit antennas.