Multi-anvil press (MAP) is a major type of apparatus for phase equilibrium, material synthesis, and property investigations at up to 120 GPa of static compression. Here I summarize the fundamental aspects of the MAP techniques, highlighting uncertainties in pressure and temperature in experiments using tungsten carbide (WC) anvils with 3 mm and 5 mm truncation edge lengths (TEL). To illustrate the application of the MAP for synthesizing millimeter-sized single crystals of deep mantle silicates, I review the theory of crystal growth and compile relevant phase diagrams. The theory informs synthesis methods such as slow cooling of a fluid solution and growing crystals in a thermal gradient. The updated phase diagrams involve Mg2SiO4, MgSiO3, and H2O at pressures up to 27 GPa and touch upon iron-bearing compositions. Also reviewed briefly are techniques to characterize the compositions and structures of synthesis products.