In this paper, the authors consider solutions of the Navier-Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. \par Of particular interest throughout this article will be the case of large Reynolds numbers, which corresponds to the limit $|\alpha|\rightarrow \infty$. For $\alpha$ large, this data falls outside the realm of the previously existing local well-posedness theory for mild solutions and, as the velocity is not in $L^2_{\rm loc}$, one cannot construct Leray-Hopf weak solutions either. \par In this article, the authors develop a framework to study general (smooth, non-self-intersecting) vortex filaments in 3d and use this to prove several existence and uniqueness results that hold for arbitrary circulation numbers $\alpha$. \par First, they prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, they prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of 3d Navier-Stokes, as well as solutions that are locally approximately self-similar.