We discuss some questions and ideas which have lead to the definition of Gevrey and Gelfand-Shilov functions. After a brief historical account for the study of such objects, we discuss different representation theorems and main properties of Gelfand-Shilov spaces and Gevrey classes. By using the approach proposed by Komatsu we extend these classes and introduce their duals, different spaces of ultra-distributions. In the second part of the talk we present some tools from time-frequency analysis and use them for the study of microlocal properties and pseudodifferential operators in the context of Gelfand-Shilov spaces, Gevrey classes and their duals.