Metamaterials are a kind of artificial microstructures, designed on the mesoscopic scale whose size is between microscopic and macroscopic scales. The equivalent material properties can be obtained from such artificial microstructures over a wide range. They also show the extraordinary characteristics such as negative equivalent mass density, negative equivalent modulus and negative refractive index. Elastic metamaterials are a new type of structural materials with extraordinary mechanical or acoustic properties. It can control elastic wave in a way that traditional materials cannot. The practical applications of elastic metamaterials are the perfect absorption of sound waves, acoustic cloak and so on. The band gap is a frequency range in which the elastic wave cannot propagate in the elastic metamaterials. We can utilize the band gap characteristic in vibration control or waveguides. In this work, the band gap of elastic metamaterials is analyzed and optimized. The main works are included as follows: Firstly, the band gap of elastic metamaterials is numerically calculated. The band gap is generally observed from the band structure of the elastic metamaterial. From the simulation software ANSYS, the complex eigenfrequency problems contained in the band structure can not be directly solved. We propose the method of real-imaginary separation to transform the complex periodic boundary conditions into the constraint equations that is easy to be solved by ANSYS. The calculation results are validated in the analysis software COMSOL. At the same time, the mode shape of local resonance elastic metamaterials is studied. It is found that the first-order band gap is mainly generated by the translational vibration of the central mass. The analysis results also show that there is an optimal unit cell configuration of elastic metamaterials, which possesses the maximal band gap. This work provides a foundation for the optimal design of band gap. Secondly, we perform optimal design to maximize the band gaps based on genetic algorithm. According to the numerical analysis results, the reasonable optimal parameters for the genetic algorithm are first obtained. The optimization of third-order relative band gap is performed as a verification example to illustrate the correctness of genetic parameters and the optimization method. Then the influence of different objective functions on the optimization results is studied. A new configuration of elastic metamaterial with wide band gap is obtained. Additionally, the excellent vibration reduction performance of the optimized structures is verified through the frequency response function. Finally, the multi-objective optimization of elastic metamaterials is carried out with the minimum weight and the maximum band gap as optimal objectives. A series of new configurations are obtained. In addition, the relationship between the mass of unit cell and the band gap is discussed. And then the multi-objective optimization of local resonance elastic metamaterials is carried out with maximizing the double band gaps. It is found that the sum of the two band gap values of the optimized configuration is basically constant. The frequency response curve is obtained. From the optimal results, a series of band gaps with different initial frequencies are obtained. Therefore, we can choose the unit cell whose initial frequency of band gap is matched with the vibration source frequency in practical applications.