A strong clique in a graph is a clique intersecting all inclusion-maximal stable sets. Strong cliques play an important role in the study of perfect graphs. We study strong cliques in the class of diamond-free graphs, from both structural and algorithmic points of view. We show that the following five NP-hard or co-NP-hard problems all remain NP-hard or co-NP-hard when restricted to the class of diamond-free graphs: Is a given clique strong? Does the graph have a strong clique? Is every vertex contained in a strong clique? Given a partition of the vertex set into cliques, is every clique in the partition strong? Can the vertex set be partitioned into strong cliques? On the positive side, we show that the following three problems whose computational complexity is open in general can be solved in polynomial time in the class of diamond-free graphs: Does every induced subgraph have a strong clique? Is every maximal clique strong? Is every edge contained in a strong clique? The last two results are derived from a characterization of diamond-free graphs in which every maximal clique is strong, which also implies an improved Erdős-Hajnal property for such graphs. •A strong clique in a graph is a clique intersecting all inclusion-maximal stable sets.•We study strong cliques in the class of diamond-free graphs.•We characterize diamond-free CIS graphs and study their Erdős-Hajnal property.•We characterize co-strongly perfect diamond-free graphs.•We derive hardness results for five related problems.