The multi-dimensional laser is a fascinating platform not only for the discovery and understanding of new higher-dimensional coherent lightwaves but also for the frontier study of the complex three-dimensional (3D) nonlineardynamics and solitary waves widely involved in physics, chemistry, biology and materials science. Systemically controlling coherent lightwave oscillation in multi-dimensional lasers, however, is challenging and has largely been unexplored;yet, it is crucial for both designing 3D coherent light fields and unveiling any underlying nonlinear complexities. Here, for the first time, we genetically harness a multi-dimensional fibre laser using intracavity wavefront shaping technology such that versatile lasing characteristics can be manipulated. We demonstrate that the output power, mode profile, optical spectrum and mode-locking operation can be genetically optimized by appropriately designing the objective function of the genetic algorithm. It is anticipated that this genetic and systematic intracavity control technology for multi-dimensional lasers will be an important step for obtaining high-performance 3D lasing and presents many possibilities for exploring multi-dimensional nonlinear dynamics and solitary waves that may enable new applications.