  In this paper, we develop a general framework of geometric functorial field
theories, meaning that all bordisms in question are endowed with geometric
structures. We take particular care to establish a notion of smooth variation
of such geometric structures, so that it makes sense to require the output of
our field theory to depend smoothly on the input. We then test our framework on
the case of $1$-dimensional field theories (with or without orientation) over a
manifold $M$. Here the expectation is that such a field theory is equivalent to
the data of a vector bundle over $M$ with connection and, in the nonoriented
case, the additional data of a nondegenerate bilinear pairing; we prove that
this is indeed the case in our framework.
