1. Bounded-shape sum-partition problems: polyhedral approach. 1.1. Linear objective: solution by LP. 1.2. Enumerating vertices of the partition polytopes and corresponding partitions using edge-directions. 1.3. Representation, characterization and enumeration of vertices of partition polytopes: distinct partitioned vectors. 1.4. Representation, characterization and enumeration of vertices of partition polytopes: general case. 2. Constrained-shape and single-size sum-partition problems: polynomial approach. 2.1. Constrained-shape partition polytopes and (almost- ) separable partitions. 2.2. Enumerating separable and limit-separable partitions of constrained-shape. 2.3. Single-size partition polytopes and cone-separable partitions. 2.4. Enumerating (limit- ) cone-separable partitions -- 3. Partitions over multi-parameter spaces: combinatorial structure. 3.1. Properties of partitions. 3.2. Counting and enumerating partition classes of single-size. 3.3. Consistency and sortability of particular partition-properties -- 4. Clustering problems over multi-parameter spaces. 4.1. Geometric properties of optimal partitions. 4.2. Geometric properties of optimal partitions for d = 2 -- 5. Sum-multipartition problems over single-parameter spaces. 5.1. Multipartitions. 5.2. Single-multishape multipartition polytopes. 5.3. Constrained-multishape multipartition polytopes. 5.4. Combinatorial properties of multipartitions. 5.5. Constrained-multishape multipartition problems with asymmetric Schur convex objective: optimization over multipartition polytopes. 5.6. Sum multipartition problems: explicit solutions -- 6. Applications. 6.1. Assembly of systems. 6.2. Group testing. 6.3. Circuit card library. 6.4. Clustering. 6.5. Abstraction of finite state machines. 6.6. Multischeduling. 6.7. Cache assignment. 6.8. The blood analyzer problem. 6.9. Joint replenishment of inventory. 6.10. Statistical hypothesis testing. 6.11. Nearest neighbor assignment. 6.12. Graph partitions. 6.13. The traveling salesman problem. 6.14. Vehicle routing. 6.15. Division of property. 6.16. The consolidation of farmland