Abstarct: Rock fracture mechanics are widely used in different fields such as stability analysis of surface and underground spaces, hydraulic fracturing, in situ stress determination, rock geothermal energy extraction and time-dependent rock failure. In rock fracture mechanics, the propagation and coalescence of cracks are studied by analytical, numerical or experimental methods. In problems with complex geometry and loadings, a numerical method such as finite element or boundary element method is commonly used. These methods encounter mesh-related difficulties in dealing with fracture mechanics problems. To overcome these difficulties, a number of meshless methods such as element free Galerkin method (EFGM) have been developed. These methods have been applied successfully in various engineering fields. The EFGM has not been used to model the rock fracture propagation and coalescence until now. In this study, the EFGM is employed to model the rock fractures and to investigate the propagation and coalescence mechanisms. In implementation of the EFGM; the linear baxse function together with the cubic spline weight function is used to construct the shape functions, Lagrange multipliers method is applied to enforce the boundary conditions, the background cell and Gauss quadrature is employed to evaluate numerical integrals and the visibility criterion is used to model the cracks. A number of examples with different specimen and crack geometry under various loads (tension, compression and shear) are considered and the stress intensity factors are calculated by different techniques. Comparison of the EFGM results with the results of finite element and analytical methods indicated that the EFGM generates accurate results for stress intensity factors. Two crack propagation criteria; baxsed on stress intensity factors and local stress components are evaluated by two examples, and the stress baxsed criterion is selected to simulate crack propagation. By this criterion the crack propagation mechanism in cubic and disk specimens with an inclined crack and the coalescence mechanism of two inclined cracks in cubic specimens for different amounts of the length and inclination of the connection line of two cracks are studied. In different stages of the EFGM modeling, the required algorithms are prepared and the computer codes are programmed in MATLAB. The length of the connection line between two cracks has no effect on the crack propagation and coalescence pattern, only by the increasing of this length, the initiation and coalescence stress increases. The inclination of the connection line has important role and controls the propagation and coalescence pattern. For different connection line inclinations, four coalescence mode including shear, shear – tension, tension – shear and tension modes are predicted by the use of numerical modeling. To verify the numerical results of EFGM, various experimental investigations are carried out. In experimental studies, pre-cracked specimens are prepared from gypsum. The mechanical properties of gypsum samples such as axial compressive strength, tension strength, elasticity module, Poison ratio, cohesion and friction angle are determined with different laboratory tests. Four set of specimens: cubic specimens containing an inclined crack, disk specimens containing an inclined crack, cubic specimens containing two inclined crack with variable connection line length and cubic specimens containing two inclined crack with variable connection line inclination are prepared and subjected to uniaxial compressive load. Qualitative and quantitative comparisons of the numerical and experimental results show that the element free Galerkin method has good accuracy and this method is capable to analyze rock fracture mechanics problems.