  We describe recent achievements in the theory of weight systems, which are
functions on chord diagrams satisfying so-called $4$-term relations. Our main
attention is devoted to constructions of weight systems. The two main sources
of these constructions that are discussed in the paper are invariants of
intersection graphs of chord diagrams that satisfy $4$-term relations for
graphs and metrized Lie algebras.
  For the simplest nontrivial metrized Lie algebra $\mathfrak {sl}(2)$, we
present recent results about the explicit form of generating functions for the
values of the corresponding weight system on important families of chord
diagrams. We also explain another recent result: construction of recurrence
relations for computing the values of the $\mathfrak{gl}(N)$-weight system.
These relations are based on M. Kazarian's extension of the
$\mathfrak{gl}(N)$-weight system to arbitrary permutations.
  Certain recent papers suggest an approach to extending weight systems and
graph invariants to arbitrary embedded graphs, which is based on the study of
the corresponding Hopf algebra structures; we describe this approach. Weight
systems defined on arbitrary embedded graphs correspond to finite type
invariants of links (multicomponent knots).
