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Abstract The NANOGrav 15 yr data provide compelling evidence for a stochastic gravitational-wave (GW) background at nanohertz frequencies. The simplest model-independent approach to characterizing the frequency spectrum of this signal consists of a simple power-law fit involving two parameters: an amplitudeAand a spectral indexγ. In this Letter, we consider the next logical step beyond this minimal spectral model, allowing for arunning(i.e., logarithmic frequency dependence) of the spectral index, ${\gamma }_{\mathrm{run}}\left(f\right)=\gamma +\beta \ln \left(f/f_{\mathrm{ref}}\right)$. We fit this running-power-law (RPL) model to the NANOGrav 15 yr data and perform a Bayesian model comparison with the minimal constant-power-law (CPL) model, which results in a 95% credible interval for the parameterβconsistent with no running, $\beta \in \left(-0.80,2.96\right)$, and an inconclusive Bayes factor, $B\left(\mathrm{RPL}\mathrm{versus}\mathrm{CPL}\right)=0.69\pm 0.01$. We thus conclude that, at present, the minimal CPL model still suffices to adequately describe the NANOGrav signal; however, future data sets may well lead to a measurement of nonzeroβ. Finally, we interpret the RPL model as a description of primordial GWs generated during cosmic inflation, which allows us to combine our results with upper limits from Big Bang nucleosynthesis, the cosmic microwave background, and LIGO–Virgo–KAGRA.