We present the identification of 34 likely binary central stars (CSs) of planetary nebulae (PNe) from Kepler/K2 data, seven of which show eclipses. Of these, 29 are new discoveries. Two additional CSs with more complicated variability are also presented. We examined the light curves of all 'possible', 'likely', and 'true' PNe in every Kepler/K2 campaign (0 through 19) to identify CS variability that may indicate a binary CS. For Campaigns 0, 2, 7, 15, and 16, we find 6 likely or confirmed variables among 21 PNe. Our primary effort, though, was focused on Campaign 11 which targeted a Galactic bulge field containing approximately 183 PNe, in which we identified 30 candidate variable CSs. The periods of these variables range from 2.3 h to 30 d, and based on our analysis, most are likely to be close binary star systems. We present periods and preliminary classifications (eclipsing, double degenerate, or irradiated systems) for the likely binaries based on light-curve shape. From our total sample of 204 target PNe, with a correction for incompleteness due to magnitude limits, we calculate a binary fraction of PN central stars to be 20.7 per cent for all the observed PNe, or 23.5 per cent if we limit our sample only to 'true' PNe. However, these fractions are almost certainly lower limits due to the large angular size of the Kepler pixels, which leads to reduced sensitivity in detecting variability, primarily as a result of dilution and noise from the nebula and neighbouring stars. We discuss the binary population of CSs based on these results as part of the total known sample of close binary CSs.