A group scheme is a group object in the category of schemes. Similarly, if is a scheme, a group scheme over is a group object in the category of schemes over .

As usual with schemes, the points of a group scheme are not the whole story. For example, a group scheme may have only one point over its field of definition and yet not be trivial. The points of the underlying topological space do not form a group under the obvious choice for a group law.

We can view a group scheme as a “group machine”: given a ring , the set of -points of forms a group. If is a scheme that is not affine, we can nevertheless interpret as a family of groups fibred over .