  Let $Q\to M$ be a principal $G$-bundle, and $B_0$ a connection on $Q$. We
introduce an infinitesimal homogeneity condition for sections in an associated
vector bundle $Q\times_GV$ with respect to $B_0$, and, inspired by the well
known Ambrose-Singer theorem, we prove the existence of a connection which
satisfies a system of parallelism conditions. We explain how this general
theorem can be used to prove the known Ambrose-Singer type theorems by an
appropriate choice of the initial system of data.We also obtain new
applications, which cannot be obtained using the known formalisms, e.g. a
classification theorem for locally homogeneous spinors.
  Finally we introduce natural local homogeneity and local symmetry conditions
for triples $(g,P\stackrel{p}{\to} M,A)$ consisting of a Riemannian metric on
$M$, a principal bundle on $M$, and a connection on $P$. Our main results
concern locally homogeneous and locally symmetric triples, and they can be
viewed as bundle versions of the Ambrose-Singer and Cartan theorem.
