The main concern of this book is the application of infinitemoment theory to the problem of controllability of one-di-mensional vibrating systems (like strings and beams) andheating processes. Distributed as well as boundary controlis considered.In the case of vibrating systems trigonometric moment pro-blems are to be investigated which is done on the basis ofan abstract moment theory in Hilbert spaces. Equivalently,alsothe theory of linear operator equations on Hilbert spa-ces ( partly with unbounded operators) is applied to theproblem of controllability and time-minimal controllability.In the case of heating processes exponential moment problemsare to be dealt with which is done on the basis of an ab-stract moment theory in Banach spaces. Time-minimal control-lability is also treated with the aid of the theory of line-aroperator equations on Banach spaces.Some advanced knowledge in functional analyis and on partialdifferential equations is preassumed for a fluent reader ofthe book, but parts of it are also readable with basic ma-thematical knowledge (for instance, the finite-dimensionalpart of the introduction).