1. Introduction  
2. Preliminaries and Notations  
3. Inverse Regression Methods  
3.1. Sliced Inverse Regression (SIR)  
3.2. Sliced Average Variance Estimation (SAVE)  
3.3. Parametric Inverse Regression (PIR)  
3.4. Contour Regression (CR)  
3.5. Directional Regression (DR)  
3.6. Likelihood-based Methods  
3.7. Graphical Regression  
4. Forward Regression Methods  
4.1. Principal Hessian Directions (PHD)  
4.2. Minimum Average Variance Estimation (MAVE)  
4.3. Conditional Variance Estimation (CVE)  
4.4. Deep Sufficient Dimension Reduction  
5. Kernel Dimension Reduction  
5.1. Supervised Kernel Dimension Reduction  
5.2. Supervised KDR for Nonlinear Regression  
5.3. Unsupervised Kernel Dimension Reduction  
6. Conclusion