In a unified approach, this paper investigates the asymptotic performance, or equivalently the large-system properties, of various point-to-point systems with antenna-array-based multiple-input multiple-output (MIMO) channels having spatial correlations at both the transmitter and the receiver. Using the replica method originally developed for statistical physics, we provide closed-form solutions to the input-output mutual information of MIMO systems: 1) at the transmit side, with either Gaussian inputs or various nonGaussian arbitrary inputs, and 2) at the receive side, with either the optimum space-time joint-decoding, or various suboptimum spatial equalizers followed by a bank of temporal decoders. Among all these combinations, in Part I, we focus on the performance analysis of the MIMO systems with the optimum space-time joint-decoding, i.e., the information-theoretic optimum decoding. Based on the framework developed in this paper, some known performance results are recovered and some new performance results are discovered. Furthermore, we provide a novel water-filling algorithm to determine the data-rate-maximizing transmit signal covariance matrix when only the channel long-term spatial correlations are available at the transmitter. In Part II, we focus on the performance analysis of the MIMO systems with various suboptimum spatial equalizers, each followed by a bank of temporal decoders.