We present two models for the space of ``long knots'' in a
manifold with boundary.  The first model is defined as a mapping space.
The second model is cosimplicial.  Both are built using
Fulton-MacPherson compactifications of configuration spaces.
These models are homotopy equivalent to the
knot spaces when the dimension of the ambient manifold is greater than
three.  When the dimension is equal to three, these models give rise to
knot invariants, which we conjecture to consist of all finite-type
invariants.  Evidence for this conjecture is given by a cohomology
spectral sequence defined by filtering these models, whose combinatorics
along the vanishing line when the ambient manifold is a Euclidean space is
equivalent to that of Vassiliev's spectral sequence.