Study of the poroelastic media is of importance in many engineering problems. Saturated porous media consist of an elastic solid skeleton with interconnected pores and the fluid fills the pores completely. Owing to the interaction of the solid skeleton and the pore fluid, the behaviour of biphasic saturated medium is completely different from the single phase continuum. The governing equations of porous media consist of constitutive relations, mass balance and balance of momentum. There are different approaches to describe the behaviour of porous media. In this dissertation the Biot theory is employed. Due to complexity of the governing equations in both quasi static and dynamic manner, finding analytical solution is impossible for many problems; so it is necessary to use the numerical methods. In this dissertation local exponential basis functions (LEBFs) are used to solve the problems of consolidation and wave propagation in saturated porous media. At the first step of this method, the domain of the problem is discretized by a grid of scattered nodes. A cloud consisting of some neibouring nodes is constructed at each node of the grid. Then the solution of the problem over each cloud is split into homogeneous and particular parts. The homogeneous part is approximated by a linear combination of exponential functions and the particular part is simply achived by a point collocation scheme. The governing field equations in both quasi static and dynamic manner are time dependant. In this dissertation Laplace transform is employed to transform the equations from time domain into Laplace domain. The problem is then solved by the use of local exponential basis functions. In the next step, the achieved results are transformed to the time domain by the use of Durbin method. The efficiency of the LEBFs in the solution of problems with porous media is assessed by solving some benchmark problems. Key Words Saturated porous media, Biot theory, Laplace domain, Meshless methods, Local exponential basis functions.