Given two Banach spaces X and Y, we introduce and study a concept of norm-attainment in the space of nuclear operators \(\mathcal N(X,Y)\) and in the projective tensor product space \(X\widehat{\otimes }_\pi Y\). We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in \(\mathcal N(X,Y)\) and in \(X\widehat{\otimes }_\pi Y\) is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in \(X\widehat{\otimes }_\pi Y\) which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.