The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by ${\displaystyle k}$, ${\displaystyle \lambda }$, or ${\displaystyle \kappa }$ and in SI units is measured in W·m⁻¹·K⁻¹. In such units, it is the amount of joules per second of thermal energy that flow per degree Kelvin (or Celsius) difference per meter of separation.

Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal conductivity. For instance, metals typically have high thermal conductivity and are very efficient at conducting heat, while the opposite is true for insulating materials such as mineral wool or Styrofoam. Metals have this high thermal conductivity due to free electrons facilitating heat transfer. Correspondingly, materials of high thermal conductivity are widely used in heat sink applications, and materials of low thermal conductivity are used as thermal insulation. The reciprocal of thermal conductivity is called thermal resistivity.

The defining equation for thermal conductivity is ${\displaystyle \mathbf {q} =-k\nabla T}$, where ${\displaystyle \mathbf {q} }$ is the heat flux, ${\displaystyle k}$ is the thermal conductivity, and ${\displaystyle \nabla T}$ is the temperature gradient. This is known as Fourier's law for heat conduction. Although commonly expressed as a scalar, the most general form of thermal conductivity is a second-rank tensor. However, the tensorial description only becomes necessary in materials which are anisotropic.

Consider a solid material placed between two environments of different temperatures. Let ${\displaystyle T_{1}}$ be the temperature at ${\displaystyle x=0}$ and ${\displaystyle T_{2}}$ be the temperature at ${\displaystyle x=L}$, and suppose ${\displaystyle T_{2}>T_{1}}$. An example of this scenario is a building on a cold winter day; the solid material in this case is the building wall, separating the cold outdoor environment from the warm indoor environment.

According to the second law of thermodynamics, heat will flow from the hot environment to the cold one as the temperature difference is equalized by diffusion. This is quantified in terms of a heat flux ${\displaystyle q}$, which gives the rate, per unit area, at which heat flows in a given direction (in this case minus x-direction). In many materials, ${\displaystyle q}$ is observed to be directly proportional to the temperature difference and inversely proportional to the separation distance ${\displaystyle L}$:

The constant of proportionality ${\displaystyle k}$ is the thermal conductivity; it is a physical property of the material. In the present scenario, since ${\displaystyle T_{2}>T_{1}}$ heat flows in the minus x-direction and ${\displaystyle q}$ is negative, which in turn means that ${\displaystyle k>0}$. In general, ${\displaystyle k}$ is always defined to be positive. The same definition of ${\displaystyle k}$ can also be extended to gases and liquids, provided other modes of energy transport, such as convection and radiation, are eliminated or accounted for.

The preceding derivation assumes that the ${\displaystyle k}$ does not change significantly as temperature is varied from ${\displaystyle T_{1}}$ to ${\displaystyle T_{2}}$. Cases in which the temperature variation of ${\displaystyle k}$ is non-negligible must be addressed using the more general definition of ${\displaystyle k}$ discussed below.

Thermal conduction is defined as the transport of energy due to random molecular motion across a temperature gradient. It is distinguished from energy transport by convection and molecular work in that it does not involve macroscopic flows or work-performing internal stresses.