1. Introduction  
2. Preliminaries  
3. Early Results: Bounds Involving Hilbert Coefficients  
4. Multiplicities  
5. Normalization of Ideals and Rings  
6. Va Sconce Los Rings and Modules  
7. The Chern Coefficients  
8. Euler Characteristics and $j$-Transforms  
9. Variation of Hilbert Coefficients  
10. Sally Modules and Special Fibers  
11. Epilogue