<p>We present a T-splines mesh generation method for complex geometries, enabling good local mesh refinement in isogeometric analysis (IGA), with the continuity and smoothness desired. Good local mesh refinement enhances the IGA computational efficiency by having a higher density of control points only in places and directions where we need higher refinement. It also enhances computational robustness by evading high-aspect-ratio elements associated with directional refinement of structured meshes. The method is based on converting a multipatch NURBS mesh created by a complex-geometry mesh generation method to a product-form-T-splines mesh, with the desired continuity and smoothness across the patch boundaries. Even just the NURBS-based complex-geometry IGA mesh generation has been a challenge, which has largely been addressed with the Complex-Geometry IGA Mesh Generation (CGIMG) and NURBS Surface-to-Volume Guided Mesh Generation (NSVGMG) methods. The method we are presenting here, Complex-Geometry T-Splines Mesh Generation (CGTSMG), is significantly advancing the state-of-the-art complex-geometry mesh generation from where the CGIMG and NSVGMG brought it. The CGTSMG can, of course, use as input a multipatch NURBS mesh created by the CGIMG or NSVGMG. After the conversion to a T-splines mesh with the desired continuity and smoothness, the mesh quality can further be improved with a good mesh relaxation method like the Fiber-Reinforced Hyperelasticity Mesh Update Method (FRHEMUM). We describe the steps involved in the CGTSMG, followed by the FRHEMUM step, and show that good mesh qualities can be achieved with this mesh generation process.</p>