  The objects of the present study are one-parameter semigroups generated by
Schr\"odinger operators with fairly general electromagnetic potentials. More
precisely, we allow scalar potentials from the Kato class and impose on the
vector potentials only local Kato-like conditions. The configuration space is
supposed to be an arbitrary open subset of multi-dimensional Euclidean space;
in case that it is a proper subset, the Schr\"odinger operator is rendered
symmetric by imposing Dirichlet boundary conditions. We discuss the continuity
of the image functions of the semigroup and show local-norm-continuity of the
semigroup in the potentials. Finally, we prove that the semigroup has a
continuous integral kernel given by a Brownian-bridge expectation. Altogether,
the article is meant to extend some of the results in B. Simon's landmark paper
[Bull. Amer. Math. Soc. (N.S.) {\bf 7}, 447--526 (1982)] to non-zero vector
potentials and more general configuration spaces.
