Design of new materials that are durable, lightweight, and environmentally sustainable are commonly inspired by natural materials and composites. However, imagine if we can design materials in a similar way that structural engineers design strong and sustainable architected structures. In fact, we are no longer limited to the natural material patterns; we can design our own complex architected materials that may perform even better than the ones that already exist in nature. Due to the recent advances in additive manufacturing or 3D printing, fabrication of such complex architected materials is becoming possible at different length scales. The objective of this talk to present a new and novel types of architected materials where their mechanical and physical properties are controlled by their internal topology or architecture and not only by their composition. The architecture of these materials and composites are based on the mathematical triply periodic minimal surfaces (TPMS) that allow us to design new types of multifunctional materials and composites with superior mechanical and physical properties. Metallic and polymeric based architected TPMS materials and composites are fabricated using macro and micro 3D printing technologies depending on the length scale of interest and desirable properties. Furthermore, the fabrication of graphene-based TPMS lattices and their mechanical and physical properties are investigated. Analytical and computational models are used to develop insights about the obtained properties and guide their further developments. Potential applications of these material and composites are demonstrated in automotive, aerospace and defense (lightweight and damage-tolerant materials), the energy (energy storage, heat transfer), semiconductor (heat dissipation), and oil and gas (catalytic supports) sectors.