Training design for general multiple-input multiple-output (MIMO) systems is investigated in this paper. Unlike prior designs that are applicable only for centralized MIMO systems with total power constraints, general power constraints are considered here. They cover total power constraints, individual power constraints, and mixed individual and per-user sum-power constraints as special cases. By writing the MIMO received signals in matrix and vector forms, respectively, and using Kronecker structured channel and noise statistics, three channel estimation schemes, i.e., right estimation, left estimation, and right-left estimation, are discussed. Their corresponding training designs are considered individually with the general power constraints. Under each channel estimation scheme, optimal training sequences to maximize the mutual information between the true channel and its estimated counterpart, and to minimize the mean square error (MSE) of the channel estimate are, respectively, proposed in semiclosed forms. The relationship between the two design criteria, i.e., the mutual information maximization and the MSE minimization, is clearly revealed. The optimal training designs under the three estimation schemes are also compared in depth. It is demonstrated that right estimation exploits less statistical information about the channel and noise, and provides worse performance than the left estimation but with lower computational complexity. On the other hand, right-left estimation performs in between the other two and provides a good compromise between complexity and performance. Finally, the optimality and effectiveness of the proposed training designs are verified by extensive simulations.