The Euclidean distance degree of an algebraic variety is a well-studied topic in applied algebra and geometry, with direct applications in geometric modeling, computer vision, and statistics. I will first describe a new topological interpretation of the Euclidean distance degree of an affine variety in terms of weighted Euler characteristics. As a concrete application, I will present a solution to the open problem in computer vision of determining the Euclidean distance degree of the affine multiview variety. Secondly, I will present a solution to a conjecture of Aluffi-Harris concerning the Euclidean distance degree of projective varieties. Projective varieties appear naturally in low rank matrix approximation, formation shape control, and all across algebraic statistics. (Joint work with J. Rodriguez and B. Wang.)