While the uptake of data-driven approaches for materials science and chemistry is at an exciting, early stage, to realise the true potential of machine learning models for successful scientific discovery, they must have qualities beyond purely predictive power. The predictions and inner workings of models should provide a certain degree of explainability by human experts, permitting the identification of potential model issues or limitations, building trust on model predictions and unveiling unexpected correlations that may lead to scientific insights. In this work, we summarize applications of interpretability and explainability techniques for materials science and chemistry and discuss how these techniques can improve the outcome of scientific studies. We discuss various challenges for interpretable machine learning in materials science and, more broadly, in scientific settings. In particular, we emphasize the risks of inferring causation or reaching generalization by purely interpreting machine learning models and the need of uncertainty estimates for model explanations. Finally, we showcase a number of exciting developments in other fields that could benefit interpretability in material science and chemistry problems.