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Space-time block codes (STBCs) from orthogonal designs (ODs) and coordinate interleaved orthogonal designs (CIOD) have been attracting wider attention due to their amenability for fast (single-symbol) maximum-likelihood (ML) decoding, and full-rate with full-rank over quasi-static fading channels. However, these codes are instances of single-symbol decodable codes and it is natural to ask, if there exist codes other than STBCs form ODs and CIODs that allow single-symbol decoding? In this paper, the above question is answered in the affirmative by characterizing all linear STBCs, that allow single-symbol ML decoding (not necessarily full-diversity) over quasi-static fading channels-calling them single-symbol decodable designs (SDD). The class SDD includes ODs and CIODs as proper subclasses. Further, among the SDD, a class of those that offer full-diversity, called Full-rank SDD (FSDD) are characterized and classified. We then concentrate on square designs and derive the maximal rate for square FSDDs using a constructional proof. It follows that 1) except for N=2, square complex ODs are not maximal rate and 2) a rate one square FSDD exist only for two and four transmit antennas. For nonsquare designs, generalized coordinate-interleaved orthogonal designs (a superset of CIODs) are presented and analyzed. Finally, for rapid-fading channels an equivalent matrix channel representation is developed, which allows the results of quasi-static fading channels to be applied to rapid-fading channels. Using this representation we show that for rapid-fading channels the rate of single-symbol decodable STBCs are independent of the number of transmit antennas and inversely proportional to the block-length of the code. Significantly, the CIOD for two transmit antennas is the only STBC that is single-symbol decodable over both quasi-static and rapid-fading channels.