In this paper we use Kuperberg's $\mathfrak{ sl}_3$-webs and Khovanov's $\mathfrak{ sl}_3$-foams to define a new algebra $K^S$, which we call the $\mathfrak{ sl}_3$-web algebra. It is the $\mathfrak{ sl}_3$ analogue of Khovanov's arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$-skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{ \oplus}_0(\mathcal{ W}^S)_{ \mathbb{ Q}(q)}$, to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra.
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