Differential forms are used in geometric measure theory to define currents. One use of currents is as a generalization of submanifolds, with better compactness properties. That is, it's easier to show that a subsequence of currents converge to a current. This can give one approach to the minimal surface problem; see this encyclopediaofmath.org article.

Maybe someone else can give some better detail or more uses of currents.

Differential forms are used in geometric measure theory to define currents. One use of currents is as a generalization of submanifolds, with better compactness properties. That is, it's easier to show that a subsequence of currents converge to a current. This can give one approach to the minimal surface problem; see the encyclopediaofmath.org article on currents.

Maybe someone else can give some better detail or more uses of currents.