This lecture will first present a brief introduction to structural topology design optimization. By this quite novel method, the shape of external as well as internal boundaries and the number of inner holes of a structure can be simultaneously optimized with respect to a predefined design objective for a prescribed total structural volume and some other constraints. Problems of topology optimization of continuum structures against vibration and noise subject to external excitation will be then considered. A frequent goal of the design of vibrating structures is to avoid resonance of the structure in a given interval for external excitation frequencies. This can be achieved by, e.g., maximizing (i) the fundamental eigenfrequency, (ii) an eigenfrequency of higher order, or (iii) the gap between two consecutive eigenfrequencies of given order. Such problems are often complicated by the fact that the eigenfrequencies in question may be multiple - or become that during the computational procedure - and this is particularly the case in topology optimization where the design freedom is generally very large. In the present paper, different approaches are considered and discussed for topology optimization involving simple and multiple eigenfrequencies of linearly elastic structures without damping. The mathematical formulations of these topology optimization problems and several illustrative results will be presented.