This article shows how the method of instantaneous distributions can be used to model the microstructures of polymers made under different polymerization conditions. The three main distributions investigated are the distributions of chain length (CLD), chemical or comonomer composition (CCD), and long chain branching (LCBD). It is also explained how the method of instantaneous distributions can be combined with reactor models to calculate the cumulative distribution of polymers made in reactors having different residence time distributions, spatial and time gradients. Finally, the usefulness of this mathematical modeling technique is illustrated in several case studies involving olefin polymerization. Extensions for free-radical polymerization are covered in the appendices. 236 www.MaterialsViews.com P 0 , monomer-free active site; P r , living chain of length r; D r , dead chain of length r; M, monomer; k p , propagation rate constant; k tH , transfer to hydrogen rate constant.Figure 10. Dependency of chain length and LCB averages on the parameters of Equation (44): a) weight average chain length, b) polydispersity index, c) LCB/1 000 C, and d) average number of LCBs per chain. Figure C1. Chemical composition distributions of chains with different chain lengths for a random terpolymer with F 1 ¼ F 2 ¼ F 3 ¼ 1=3, and m 1 ¼ 28, m 2 ¼ 42, and m 3 ¼ 56: a) r ¼ 100, b) r ¼ 250, and c) r ¼ 500.