The aim of this review is to give an overview of techniques and methods used in the modeling of acoustic and elastic metamaterials. Acoustic and elastic metamaterials are man-made materials which present exotic properties capable to modify and drive wave propagation. In particular in this work we will focus on locally resonant microstructures. Such metamaterials are based on local resonances of the internal structure, the dimensions of which are much smaller than the wavelengths of the waves under analysis. We will consider the seminal papers in the fields to grasp the most important ideas used to develop locally resonant metamaterials, such as homogenization techniques and optimization topology. Finally, we will discuss some interesting application to clarify the aforementioned methods.