Robustness means clearly stating assumptions your methods and data rely on, and gradually relaxing each of them to see if your results still hold. It acts as an efficient coherence check if you realize your findings can dramatically change due to a single variable, especially if that variable is subject to noise, error measurement, etc.
Direct Acyclic Graphs (DAGs) are a great tool for checking robustness. They help you clearly spell out assumptions and hypotheses in the context of causal inference.
Dagitty, is a handy browser-based tool. In a nutshell, when you draw an assumed chain of causal events in Dagitty, it provides you with robustness checks on your data, like certain conditional correlations that should vanish.