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The thermodynamic structure of hot gas in galaxy clusters is sensitive to astrophysical processes and typically difficult to model with galaxy formation simulations. We explore the fraction of cool-core (CC) clusters in a large sample of 370 clusters from IllustrisTNG, examining six common CC definitions. IllustrisTNG produces continuous CC criteria distributions, the extremes of which are classified as CC and non-cool core (NCC), and the criteria are increasingly correlated for more massive clusters. At = 0, the CC fractions for two criteria are in reasonable agreement with the observed fractions but the other four CC fractions are lower than observed. This result is partly driven by systematic differences between the simulated and observed gas fraction profiles. The simulated CC fractions with redshift show tentative agreement with the observed fractions, but linear fits demonstrate that the simulated evolution is …