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We describe recent achievements in the theory of weight systems, which arefunctions on chord diagrams satisfying so-called $4$-term relations. Our mainattention is devoted to constructions of weight systems. The two main sourcesof these constructions that are discussed in the paper are invariants ofintersection graphs of chord diagrams that satisfy $4$-term relations forgraphs and metrized Lie algebras.  For the simplest nontrivial metrized Lie algebra $\mathfrak {sl}(2)$, wepresent recent results about the explicit form of generating functions for thevalues of the corresponding weight system on important families of chorddiagrams. We also explain another recent result: construction of recurrencerelations 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 andgraph invariants to arbitrary embedded graphs, which is based on the study ofthe corresponding Hopf algebra structures; we describe this approach. Weightsystems defined on arbitrary embedded graphs correspond to finite typeinvariants of links (multicomponent knots).

Weight systems and invariants of graphs and embedded graphs