Summary: On finite structures, there is a well-known connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates. If the template \(\Gamma \) is \(\omega \)-categorical, we present various equivalent characterizations of those \(\Gamma \) such that the constraint satisfaction problem (CSP) for \(\Gamma \) can be solved by a Datalog program. We also show that CSP\((\Gamma )\) can be solved in polynomial time for arbitrary \(\omega \)-categorical structures \(\Gamma \) if the input is restricted to instances of bounded treewidth. Finally, we characterize those \(\omega \)-categorical templates whose CSP has Datalog width 1, and those whose CSP has strict Datalog width \(k\).