Let \(X \subset \mathbb{P}^N\) be a smooth irreducible n-dimensional variety. A well-known conjecture predicts that \(X\) always carries an Ulrich vector bundle, that is a bundle \(\mathcal{E}\) such that \(H^i(\mathcal{E}(−p)) = 0\) for \(i \geq 0\) and \(1 \leq p \leq n\). In the talk we will report on three recent results in collaboration with D. Raychaudhury. The first one is that any given \(X\) carries an Ulrich bundle if and only if it contains a subvariety satisfying certain conditions. The second one is an application of this result to low rank Ulrich bundles on complete intersections of dimension \(n ≥ 5\), or on general complete intersections of dimension \(n = 4\). The third one is an application to rank 2 Ulrich bundles on general hypersurfaces of dimension \(n\) with \(2 \leq n \leq 3\).