A charged particle beam will rapidly ionize any residual gas in an accelerator's vacuum chamber, and, if that beam is negatively charged, the resulting positive ions can become trapped within the beam. Ion trapping has often been observed in circular accelerators, but has never before been seen in single-pass linear accelerators. However a new class of high intensity linacs will be the first such linear accelerators that experience ion trapping. In the Cornell photoinjector, we have recently observed this phenomenon for the first time, and we will share our experiences conducting experiments to study ion trapping, as well as theory and simulations modeling the phenomenon. We start by outlining theories to determine whether or not ion trapping will occur in an accelerator. We describe in detail some of the effects that ions can have on a beam, including emittance growth, optical errors, beam losses and even beam instabilities. The severity of these effects varies widely depending on the accelerator in question, so we offer up several simulation techniques that can be used to predict their occurrence, as well as ion signatures that can be observed experimentally to confirm the presence of ions. We share results from experiments that tested three major ion clearing methods: ion clearing electrodes, bunch gaps, and beam shaking. Results obtained from these experiments are supported by various theories and simulation codes. Finally, because taking beam property measurements in the regime where ion trapping occurs can be difficult, we offer up a new design for a rotating wire scanner capable of obtaining beam profiles at high beam intensity.