In this paper, we generalize the concept of unbounded norm (un) convergence: let X be a normed lattice and Y a vector lattice such that X is an order dense ideal in Y; we say that a net $$(y_\alpha )$$ (yα) un-converges to y in Y with respect to X if $$\bigl |\bigl ||y_\alpha -y|\wedge x\bigr |\bigr |\rightarrow 0$$ |||yα-y|∧x||→0 for every $$x\in X_+$$ x∈X+ . We extend several known results about un-convergence and un-topology to this new setting. We consider the special case when Y is the universal completion of X. If $$Y=L_0(\mu )$$ Y=L0(μ) , the space of all $$\mu $$ μ -measurable functions, and X is an order continuous Banach function space in Y, then the un-convergence on Y agrees with the convergence in measure. If X is atomic and order complete and $$Y=\mathbb R^A$$ Y=RA then the un-convergence on Y agrees with the coordinate-wise convergence.