  We review a cochain-free treatment of the classical van Kampen obstruction
\theta to embeddability of an n-polyhedron into R^{2n} and consider several
analogues and generalizations of \theta, including an extraordinary lift of
\theta which in the manifold case has been studied by J.-P. Dax. The following
results are obtained.
  - The mod 2 reduction of \theta is incomplete, which answers a question of
Sarkaria.
  - An odd-dimensional analogue of \theta is a complete obstruction to linkless
embeddability (="intrinsic unlinking") of the given n-polyhedron in R^{2n+1}.
  - A "blown up" 1-parameter version of \theta is a universal type 1 invariant
of singular knots, i.e. knots in R^3 with a finite number of rigid transverse
double points. We use it to decide in simple homological terms when a given
integer-valued type 1 invariant of singular knots admits an integral arrow
diagram (=Polyak-Viro) formula.
  - Settling a problem of Yashchenko in the metastable range, we obtain that
every PL manifold N, non-embeddable in a given R^m, m\ge 3(n+1)/2, contains a
subset X such that no map N\to R^m sends X and N-X to disjoint sets.
  - We elaborate on McCrory's analysis of the Zeeman spectral sequence to
geometrically characterize "k-co-connected and locally k-co-connected"
polyhedra, which we embed in R^{2n-k} for k<(n-3)/2 extending the
Penrose-Whitehead-Zeeman theorem.
