When the central bank has information that can help the private sector better predict the future, should it communicate such information to the public? In purely forward-looking New Keynesian models, such Delphic forward guidance unambiguously reduces ex ante welfare by increasing the variability of inflation and the output gap. We call this phenomenon the Delphic forward guidance puzzle. In more elaborate models with endogenous state variables, a combination of Delphic forward guidance and preemptive policy actions may improve welfare. However, full information revelation is generally not optimal and what information needs to be revealed is highly model-dependent.