Numerical climate models (or climate system models) are mathematical models that can simulate the interactions of important drivers of climate. These drivers are the atmosphere, oceans, land surface and ice. Scientists use climate models to study the dynamics of the climate system and to make projections of future climate and of climate change. Climate models can also be qualitative (i.e. not numerical) models and contain narratives, largely descriptive, of possible futures.

Climate models take account of incoming energy from the Sun as well as outgoing energy from Earth. An imbalance results in a change in temperature. The incoming energy from the Sun is in the form of short wave electromagnetic radiation, chiefly visible and short-wave (near) infrared. The outgoing energy is in the form of long wave (far) infrared electromagnetic energy. These processes are part of the greenhouse effect.

Climate models vary in complexity. For example, a simple radiant heat transfer model treats the Earth as a single point and averages outgoing energy. This can be expanded vertically (radiative-convective models) and horizontally. More complex models are the coupled atmosphere–ocean–sea ice global climate models. These types of models solve the full equations for mass transfer, energy transfer and radiant exchange. In addition, other types of models can be interlinked. For example Earth System Models include also land use as well as land use changes. This allows researchers to predict the interactions between climate and ecosystems.

Climate models are systems of differential equations based on the basic laws of physics, fluid motion, and chemistry. Scientists divide the planet into a 3-dimensional grid and apply the basic equations to those grids. Atmospheric models calculate winds, heat transfer, radiation, relative humidity, and surface hydrology within each grid and evaluate interactions with neighboring points. These are coupled with oceanic models to simulate climate variability and change that occurs on different timescales due to shifting ocean currents and the much larger heat storage capacity of the global ocean. External drivers of change may also be applied. Including an ice-sheet model better accounts for long term effects such as sea level rise.

Complex climate models enable extreme event attribution, which is the science of identifying and quantifying the role that human-caused climate change plays in the frequency, intensity and impacts of extreme weather events. Attribution science aims to determine the degree to which such events can be explained by or linked to human-caused global warming, and are not simply due to random climate variability or natural weather patterns.

There are three major types of institution where climate models are developed, implemented and used:

Big climate models are essential but they are not perfect. Attention still needs to be given to the real world (what is happening and why). The global models are essential to assimilate all the observations, especially from space (satellites) and produce comprehensive analyses of what is happening, and then they can be used to make predictions/projections. Simple models have a role to play that is widely abused and fails to recognize the simplifications such as not including a water cycle.

A general circulation model (GCM) is a type of climate model. It employs a mathematical model of the general circulation of a planetary atmosphere or ocean. It uses the Navier–Stokes equations on a rotating sphere with thermodynamic terms for various energy sources (radiation, latent heat). These equations are the basis for computer programs used to simulate the Earth's atmosphere or oceans. Atmospheric and oceanic GCMs (AGCM and OGCM) are key components along with sea ice and land-surface components.