Applied harmonic analysis plays an important role in engineering such as signal processing, image processing, digital communications, medical imaging and so on. This study has present an overview of all kind of reproducing systems which are obtained by applying a combination of dilations, modulations and translations to a finite family of functions. Five reproducing systems and their applications are mainly introduced as the following: Gabor systems, wavelet systems, wave packet systems, composite dilation wavelet systems and shearlet systems. Present study reviewed their definitions, history and existing known results, respectively. Furthermore, it also discussed their advantages and shortcomings in the engineering applications.