  Cosmic filaments are the main transport channels of matter in the Megaparsec
universe, and represent the most prominent structural feature in the matter and
galaxy distribution.
  Here we describe and define the physical and dynamical nature of cosmic
filaments. It is based on the realization that the complex spatial pattern and
connectivity of the cosmic web are already visible in the primordial random
Gaussian density field, in the spatial pattern of the primordial tidal and
deformation eigenvalue field. The filaments and other structural features in
the cosmic web emerging from this are multistream features and structural
singularities in phase-space. The caustic skeleton formalism allows a fully
analytical classification, identification, and treatment of the nonlinear
cosmic web. The caustic conditions yield the mathematical specification of
weblike structures in terms of the primordial deformation tensor eigenvalue and
eigenvector fields, in which filaments are identified -- in 2D -- with the
so-called cusp caustics. These are centered around points that are maximally
stretched as a result of the tidal force field.
  The resulting mathematical conditions represent a complete characterization
of filaments in terms of their formation history, dynamics, and orientation. We
illustrate the workings of the formalism on the basis of a set of constrained
$N$-body simulations of protofilament realizations. These realizations are
analyzed in terms of spatial structure, density profiles, and multistream
structure and compared to simpler density or potential field saddle point
specifications.
  The presented formalism, and its 3D generalization, will facilitate the
mining of the rich cosmological information contained in the observed weblike
galaxy distribution, and be of key significance for the analysis of
cosmological surveys such as SDSS, DESI, and Euclid.
