This chapter focuses on research progress related to full quantum effects in dense hydrogen, hydride compounds, and proton transport through 2D materials and bulk materials where hydrogen does not dominate in content. For dense hydrogen, this chapter describes nuclear quantum effects in high-pressure phase diagram, phase transition, and physical property. For the most common hydride compound, water, this chapter introduces nuclear quantum effects in water hydrogen bond property, water phase diagram, and physical and chemical property of confined water and ice. For other hydride compound, this chapter introduces full quantum effects in hydride superconductivity and nonlinear optical property of mineral hydride. These studies reveal that, full quantum effect not only has significant contribution to existing properties, such as hydrogen bond strength and phase diagram boundary probed by isotope substitution, but also causes novel states and properties, such as room temperature, high-pressure superconductivity of hydrides (e.g., sulfur hydride) and liquid forms of metallic and superconducting hydrogen. After these, this chapter introduces theory of proton tunneling rate, which is important from physical to life sciences. Theory of proton transport in perovskite-type oxides is also illustrated as typical example of nuclear quantum effects on hydrogen conduction in bulk materials. Finally, the chapter introduces work demonstrating importance of proton tunneling in DNA mutation.