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In this paper, we explore some of the fundamentals of synchronous Code Division Multiple Access (CDMA) as applied to wireless and optical communication systems under very general settings (of any size) for the user symbols and the signature matrix entries. The channel is modeled by real/complex additive noise of arbitrary distribution. Two problems are addressed. The first problem concerns whether uniquely detectable overloaded matrices exist in the absence of additive noise under these general settings, and if so, whether there are any practical optimum detection algorithms. The second one is about the bounds for the sum channel capacity when user data and signature matrices employ any real or complex alphabets (finite or infinite). In response to the first problem, we have developed practical maximum likelihood detection algorithms for overloaded CDMA systems for a large class of alphabets. In response to the second problem, a general theorem has been developed in which the sum capacity lower bounds with respect to the number of users, spreading gain, and signal-to-noise ratio can be derived. To show the power and utility of the main theorem, a number of sum capacity bounds for special cases are evaluated. An important conclusion of this paper is that the lower and upper bounds of the sum capacity for small/medium-size CDMA systems depend on both the input and the signature symbols; this is contrary to the asymptotic results for large-scale systems reported in the literature (also confirmed in this paper) where the signature symbols and statistics disappear for signature matrices and input vectors with i.i.d. entries. Furthermore, upper and asymptotic bounds are derived and compared to other derivations.