We study the main operations (argument shift, differentiation, etc.) in the generalized spaces of type S and some classes of analytic functions and pseudodifferential operators in spaces of this kind, as well as the properties of the Fourier transforms of generalized functions, convolutions, convolvers, and multiplicators. The correct solvability of the nonlocal time problem is proved for one class of pseudodifferential equations in generalized spaces of type S. Its solution is presented in the form of convolution of the fundamental solution with an initial function, which is an element of the space of generalized functions of the ultradistribution type.