Actively accreting supermassive black holes (SMBHs) are found, nearly universally, to create luminous X-ray emission. However, there are apparent X-ray weak exceptions to this rule that are now providing novel insights, including many weak-line quasars (WLQs). We have been systematically studying such X-ray weak quasars with Chandra observations and near-infrared spectroscopy, and I will report results on their remarkable properties and describe implications for models of the accretion disk/corona, quasar winds, and emission-line formation. We have found evidence that many of these WLQs have geometrically thick inner accretion disks, likely due to high Eddington ratios, that shield the high-ionization broad line region from the relevant ionizing continuum. This basic model can explain, in a unified manner, the weak lines and diverse X-ray properties of WLQs. Such shielding may, more generally, play a significant role in shaping the broad distributions of quasar emission-line equivalent widths. An expectation of our model is that WLQs should be more common at high redshift, and they may serve as a signature of rapid SMBH growth at early cosmic times. I will end by discussing some promising ongoing studies that are extending these ideas.