Preliminaries. Ranks and Incomplete Types. Global Theory. Prime Models. More on Types and Saturated Models. Saturation of Ultraproducts. Construction of Models. The Number of Non-Isomorphic Models in Pseudo-Elementary Classes. Categoricity and the Number of Models in Elementary Classes. Classification for FaNo-Saturated Models. The Decomposition Theorem. The Main Gap For Countable Theories. For Thomas the Doubter. Appendix: Filters, Stationary Sets and Families of Sets. Partition Theorems. Various Results. Historical Remarks. References.