A space-time block code (STBC) in K symbols (variables) is called a g-group decodable STBC if its maximum-likelihood (ML) decoding metric can be written as a sum of g terms, for some positive integer g greater than one, such that each term is a function of a subset of the K variables and each variable appears in only one term. In this paper, we provide a general structure of the weight matrices of multigroup decodable codes using Clifford algebras. Without assuming that the number of variables in each group is the same, a method of explicitly constructing the weight matrices of full-diversity, delay-optimal multigroup decodable codes is presented for arbitrary number of antennas. For the special case of 2 ^a number of transmit antennas, we construct two subclass of codes: 1) a class of 2 ^a -group decodable codes with rate [(a)/(2⁽ a-1))], which is, equivalently, a class of single-symbol decodable codes, and 2) a class of (2a-2)-group decodable codes with rate [((a-1))/(2⁽ a-2))], i.e., a class of double-symbol decodable codes.