A manifold is a mathematical concept that describes a topological space that locally resembles Euclidean space near each point. More specifically, a manifold is a topological space with a collection of coordinate charts, which are open sets with homeomorphisms to a Euclidean space, and patching functions: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate charts. In other words, a manifold is a space that can be "charted".

There are many different kinds of manifolds, including topological manifolds, smooth manifolds, and Riemannian manifolds. Topological manifolds are the simplest kind of manifold to define and look locally like some "ordinary" Euclidean space. Smooth manifolds, also called differentiable manifolds, are manifolds for which overlapping charts "relate smoothly" to each other, meaning that the inverse of one followed by the other is an infinitely differentiable map from Euclidean space to itself. Riemannian manifolds are smooth manifolds equipped with a Riemannian metric, which allows for the measurement of distances and angles on the manifold.

Manifolds are important in many areas of mathematics and physics because they allow complicated structures to be described in terms of well-understood topological properties of simpler spaces. They naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds have applications in computer graphics, robotics, and physics.