Exploiting the Lax form of the Korteweg-de Vries equation, we determine the whole set of its symmetry operators and relate them to conserved quantities. We also study a KdV supersymmetric version and put in evidence its even (bosonic) and odd (fermionic) symmetries. We then get the Lie superalgebra characterizing this supersymmetric nonlinear equation. © World Scientific Publishing Company.