Gaussian MIMO channel under total transmit and multiple interference power constraints (TPC and IPCs) is considered. A closed-form solution for its optimal transmit covariance matrix is obtained in the general case (up to dual variables). A number of more explicit closed-form solutions are obtained in some special cases, including full-rank and rank-1 (beamforming) solutions, which differ significantly from the well-known water-filling solutions (e.g. signaling on the channel eigenmodes is not optimal anymore and the capacity can be zero for non-zero transmit power). A whitening filter is shown to be an important part of optimal precoding under interference constraints. Capacity scaling with transmit power is studied: its qualitative behaviour is determined by a natural linear-algebraic structure induced by MIMO channels of multiple users. A simple rank condition is given to characterize the cases where spectrum sharing is possible. An interplay between the TPC and IPCs is investigated, including the transition from power-limited to interference-limited regimes. A number of unusual properties of an optimal covariance matrix under IPCs are pointed out and a bound on its rank is established. Partial null forming known in the adaptive antenna array literature is shown to be optimal from the information-theoretic perspective as well in some cases.