Ultra-stabilized echelle spectrographs (e.g., ESPRESSO, HPF, EXPRES, NEID) are enabling a new generation of precision radial velocity planet surveys. Thanks to improved instrumentation, the dominant source of “noise” is typically intrinsic stellar variability, rather than measurement uncertainties. This has motivated a variety of approaches to mitigate the effect of stellar variability on Doppler planet surveys. I propose to provide a survey of recent progress in developing, verifying and validating advanced statistical and machine learning methods for enabling Doppler planet surveys to pierce the veil of stellar variability. I would discuss the implications for planning future extremely precise radial velocity surveys and the potential implications for planning future observations to characterize the atmospheres and/or surfaces of potentially habitable exoplanets.