This talk is based on joint work with Weiwei Pan, Daniel Tubbenhauer and with Yasuyoshi Yonezawa. In his seminal work on the categorification of the Jones polynomial, Khovanov introduced a new family of algebras, which he called the arc algebras. He showed that the Grothendieck group of the category of f.d. representations of an arc algebras is isomorphic to a certain space of invariant quantum sl(2) intertwiners. This isomorphism is particularly nice, because it maps the dual canonical basis in the space of intertwiners to the classes of the indecomposable projective modules of the arc algebra. Stroppel and Webster showed that the arc algebras can also be obtained from the intersection cohomology of 2-row Springer varieties, a result which generalizes Khovanov's earlier result that their center is isomorphic to the ordinary cohomology of these Springer varieties. Brundan and Stroppel studied the representation theory of the arc algebras in great detail. A fundamental result is that the arc algebras are Morita equivalent to blocks of certain level-2 cyclotomic Khovanov-Lauda-Rouquier algebras, which are graded versions of the cyclotomic Hecke algebras. This Morita equivalence is based on a categorification of quantum skew Howe duality, which is a generalisation of Schur-Weyl duality. In joint work with the collaborators above I introduced the sl(n) generalisation of the arc algebras, which we called the sl(n) web algebras. In my talk I will explain their definition and sketch the sl(n) version of some the results above.