It is well understood that MIMO technology could enhance the reliability of wireless communication and increase the channel capacity. The design of space-time code to explore the benefit provided by the multi-antenna systems is of key importance. This dissertation addresses several issues concerning the design of space-time code. The following is a brief description of these issues and our contributions. Cyclic division algebra (CDA) has been introduced as a means to construct full-rate nonvanishing determinant STBC (space-time block code), which achieves the diversity-multiplexing trade-off and has a very good performance. There are two steps to construct CDA-based nonvanishing determinant STBC: construction of a cyclic extension over [Special characters omitted.] (i) and finding a non-norm element. For the first step we proposed a new up-to-down construction method. With this new method we find a broad range of cyclic extensions over [Special characters omitted.] (i), which encompasses all the previous constructions. For the second step, we give new criteria for the non-norm element. Non-norm elements found by these new criteria have smaller absolute values, hence the resulted STBC has a better coding gain. The well-known design criteria for space-time code is proposed by Guey-Fitz-Bell-Kuo in 1996 and Tarokh-Seshadri-Calderbank in 1998. The derivation of the design criteria is based on the assumption that the received signals are decoded with an ML receiver. One important issue seems to be long ignored: there is no design criterion for space-time code decoded with suboptimal receivers. Only until recently that Zhang-Liu-Wong and Shang-Xia studied the full diversity codes with linear receivers. We address the issue in a much broader sense. We proposed a more general receiver structure called PIC (partial interference cancellation) group decoding. A PIC group decoding can be viewed as an intermediate decoding algorithm between linear decoding and ML decoding. It encompasses both linear decoding and ML decoding as its two extremes. We also derived a design criterion for space-time codes with PIC receivers to achieve full diversity. The full diversity criteria for codes with ML receivers and linear receivers are special cases of our new design criterion. In many applications, wireless communication devices are limited by size or hardware complexity to one antenna. Cooperative communication was introduced for communication networks with single-antenna nodes to exploit the multi-path diversity. In cooperative communications, a few nodes positioned between the source node and destination node are served as the relay nodes. One important problem for cooperative communication networks is the time-asynchronism among the relay nodes. We propose a distributed space-time coding scheme called distributed linear convolutional space-time code (DLC-STC) to address this problem. We also give systematic construction methods of DLC-STC which achieves full diversity without time synchronization among the relay nodes. Furthermore, we show that our proposed DLC-STC achieves full diversity even with suboptimal receivers such as ZF/MMSE receiver and DFE receiver.