AbstractFor a primepand a fieldkof characteristic$p,$we define Steenrod operations$P^{n}_{k}$on motivic cohomology with$\mathbb {F}_{p}$-coefficients of smooth varieties defined over the base field$k.$We show that$P^{n}_{k}$is thepth power on$H^{2n,n}(-,\mathbb {F}_{p}) \cong CH^{n}(-)/p$and prove an instability result for the operations. Restricted to modpChow groups, we show that the operations satisfy the expected Adem relations and Cartan formula. Using these new operations, we remove previous restrictions on the characteristic of the base field for Rost’s degree formula. Over a base field of characteristic$2,$we obtain new results on quadratic forms.