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Since the realization of Bose-Einstein condensates (BECs) in optical potentials, intensive experimental and theoretical investigations have been carried out for matter-wave solitons, coherent structures, modulational instability (MI), and nonlinear excitation of BEC matter waves, making them objects of fundamental interest in the vast realm of nonlinear physics and soft condensed-matter physics. Ubiquitous models, which are relevant to the description of diverse nonlinear media are provided by the nonlinear Schrodinger (NLS), alias Gross-Pitaevskii (GP) equations. In many settings, nontrivial solitons and coherent structures, which do not exist or are unstable in free space, can be created or stabilized by means of various management techniques, which are represented by NLS and GP equations with spatiotemporal coefficients in front of linear or nonlinear terms. Developing this direction of research in various settings, efficient schemes of the spatiotemporal modulation of coefficients in the NLS/GP equations have been designed to engineer desirable robust nonlinear modes. This direction and related ones are the main topic of the present review. A broad and important theme is the creation and control of 1D solitons in BEC by means of combination of the temporal or spatial modulation of the nonlinearity strength and a time-varying trapping potential. An essential ramification of this topic is analytical and numerical analysis of MI of continuous-wave states, and control of the nonlinear development of MI. In addition to that, the review also includes some topics that do not directly include spatiotemporal modulation but address physically important phenomena which demonstrate similar soliton dynamics. These are soliton motion in binary BEC, three-component solitons in spinor BEC, and dynamics of two-component solitons under the action of spin-orbit coupling.