Gravity gradient tensor is the second derivatives of gravitational field. Compared with the traditional gravitational exploration, gravity gradient tensor can reach a better resolution. Gravity gradient tensor has 5 independent components, so they can contain more geological information. Therefore, gravity gradient tensor can be used to recover the causative bodies in the subsurface with high accuracy. Due to strong adaptability and flexibility of the unstructured grid, it can smoothly approximate the irregular and complex boundaries of anomalous source. Compared with the structured grid, the unstructured grid can provide researchers with more accurate discretization and calculation with less computation time. In order to reduce the ambiguity of the inversion, the authors chose to jointly and simultaneously invert all gravity gradient components based on tetrahedron grids with the help of the so called generalized objective function which is widely used in geophysical inverse problems. The authors applied the algorithm to each gravity tensor component at the beginning to ensure that the depth weighting function makes a difference in the inversion. It turns out that the depth weighting function works well. To explore all components of the gravity gradient tensor as much as possible, the authors described the joint-inversion in detail in this paper. The inversion results show that 3D inversion of gravity gradient tensor based on unstructured grids can obtain the position of causative bodies in the subsurface and the density distribution. Based on a comparison with rectangular grid inversion results, the authors highlighted the advantages of unstructured grid inversion. And the practicability of the method was verified through two synthetic models.