  We present boundary-integral equations for Maxwell-type problems in a
differential-form setting. Maxwell-type problems are governed by the
differential equation $(\delta\mathrm{d}-k^2)\omega = 0$, where
$k\in\mathbb{C}$ holds, subject to some restrictions. This problem class
generalizes $\textbf{curl}\,\textbf{curl}$- and
$\mathrm{div}\,\textbf{grad}$-types of problems in three dimensions. The goal
of the paper is threefold: 1) Establish the Sobolev-space framework in the full
generality of differential-form calculus on a smooth manifold of arbitrary
dimension and with Lipschitz boundary. 2) Introduce integral transformations
and fundamental solutions, and derive a representation formula for Maxwell-type
problems. 3) Leverage the power of differential-form calculus to gain insight
into properties and inherent symmetries of boundary-integral equations of
Maxwell-type.
