In least-squares approximations the normal equations act to project a vector existing in N-dimensional space onto a lower dimensional space, where our problem actually lies, thus providing the "best" solution we can hope for (the orthogonal projection of the N-vector onto our solution space). The "best" solution is the one that minimizes the Euclidean distance (two-norm) between the N-dimensional vector and our lower dimensional space.
There exist other norms and other spaces besides $\mathbb{R}^d$, what are the analogues of least-squares under a different norm, or in a different space?