These lecture notes provide a detailed treatment of the thermal energy storage and transport by conduction in natural and fabricated structures. Thermal energy in two carriers, i.e. phonons and electrons — are explored from first principles. For solid-state transport, a common Landauer framework is used for heat flow. Issues including the quantum of thermal conductance, ballistic interface resistance, and carrier scattering are elucidated. Bulk material properties, such as thermal and electrical conductivity, are derived from particle transport theories, and the effects of spatial confinement on these properties are established.