Abstarct: In this study, numerical simulation of incompressible flow inside two-dimensional non-square cavities are studied via lattice Boltzmann method. In fact, two kinds of problems are studied. First, incompressible flow inside non-square lid driven cavities including semi-ellipse, triangular and arc-square cavities are investigated and the results are presented in form of streamlines for different Reynolds numbers. Also, for triangular cavity, the effect of lid direction and for arc-square cavity, the effect of cavity dimensions are investigated. The results show that as Reynolds number increases, the number of the vortexes increases and with change of cavity dimensions and lid direction, the different flow pattern is created. Then, natural convection heat transfer inside non-square cavities including triangular, trapezoidal and semi-circular cavities are analyzed and for these cavities, the simulation was carried out with a fluid with Pr=0.71 and the streamlines and temperature profiles and local Nusselt number are presented for different Rayleigh numbers. Results indicate that the for low Rayleigh numbers, the isotherm lines are smooth and monotonic. As Rayleigh number increases, the temperature contours getting condensed. Also, for triangular cavity, the effect of situation of cavity and side angles on the heat transfer rate and flow pattern are studied. The results show that in triangular with right angle and constant height, when the right angle of the cavity is located at the lower point of cavity, with increasing side angles, the local Nusselt number at vertical wall increases. In contrast, when the right angle is at the top point of the cavity, the local Nusselt number of points near the right angle on the vertical wall decreases with increasing side angles. In fact, a minimum value is observed in the diagram of variation of local Nusselt numbers at vertical wall with distance from horizontal wall. Also, the results for trapezoidal cavity are presented for two different angles of cavity and it is shown that local Nusselt number at hot bottom wall is greater for smaller angle between sides and height of cavity. The results predicted by the LBM model are in good agreement with the numerical results presented. Considering advantages of LBM compared to traditional methods (simplicity of programming, locality of computation, natural parallelism and easiness in dealing with complex boundaries), it can be a good alternative for traditional methods.