A polynomial-time algorithm deciding satisfiability of CSP instances treats all unsatisfiable instances the same. An approximation algorithm for CSP may not distinguish satisfiable and almost satisfiable instances. A natural combination of these notions was suggested by Zwick in 1998: call an algorithm for CSP robust if, for each epsilon and each (1-epsilon)-satisiable instance, the algorithm returns a (1-g(epsilon)) satisfying assignment where g(epsilon) tends to 0 as epsilon tends to 0. Informally, such an algorithm correctly identifies satisfiable instances and also finds almost satisfying assignments for almost satisfiable instances. A classical result by Hastad rules out robust algorithms for systems of linear equations with three variables per equation. On the other hand, Zwick gave robust algorithms for 2-Sat and Horn-k-Sat. In 2010, Guruswami and Zhou conjectured that a CSP (with a fixed finite set of allowed constraint types over a fixed finite domain) admits a robust algorithm if and only if it has bounded width (i.e. satisfiability can be decided by a local propagation algorithm). In this talk, we will present some initial results towards proving this conjecture, based on joint work with Victor Dalmau.