The Standard Model (SM) explains CP violation in terms of the CKM matrix. The BABAR experiment was designed mainly to test the CKM model in B decays. B decays that proceed through b → s loop diagrams, of which B → KKK decays are an example, are sensitive to new physics effects that could lead to deviations from the CKM predictions for CP violation. We present studies of CP violation in the decays B+ → K+K-K+, B+ → KS0KS0K+, and B0 → K+K-KS0, using a Dalitz plot amplitude analysis. These studies are based on approximately 470 million B$\bar{B}$ decays collected by BABAR at the PEP-II collider at SLAC. We perform measurements of time-dependent CP violation in B0 → K+K-KS0, including B0 → [Phi]KS0. We measure a CP-violating phase [beta]eff ([Phi]KS0) = 0.36 ± 0.11 ± 0.04 rad., in agreement with the SM. This is the world's most precise measurement of this quantity. We also measure direct CP asymmetries in all three decay modes, including the direct CP asymmetry ACP ([Phi]K+) = (12.8 ± 4.4 ± 1.3)%, which is 2.8 sigma away from zero. This measurement is in tension with the SM, which predicts an asymmetry of a few percent. We also study the resonant and nonresonant features in the B → KKK Dalitz plots. We find that the hypothetical scalar fX(1500) resonance, introduced by prior analyses to explain an unknown peak in the mKK spectrum, cannot adequately describe the data. We conclude instead that the fX(1500) can be explained as the sum of the f0(1500), f'2(1525), and f0(1710) resonances, removing the need for the hypothetical fX(1500). We also find that an exponential nonresonant model, used by previous analyses to describe the broad nonresonant feature seen in B → KKK decays, cannot fully model the data. We introduce a new nonresonant model that contains more free parameters, allows for phase motion, and contains both S-wave and P-wave components.