Leaky waves have been known for many years in the context of leaky-wave antennas, but it is only within the past dozen years or so that it was realized that the dominant mode on printed-circuit transmission lines used in microwave and millimeter-wave integrated circuits can also leak. Such leakage is extremely important because it may cause power loss, cross talk between neighboring parts of the circuit, and various undesired package effects. These effects can ruin the performance of the circuit, so we must know when leakage can occur and how to avoid it. In most cases, these transmission lines leak only at high frequencies, but some lines leak at all frequencies. However, those lines can be modified to avoid the leakage. This paper explains why and when leakage occurs, and shows how the dominant mode behaves on different lines. The paper also examines certain less well known but important features involving unexpected new physical effects. These include an additional dominant mode on microstrip line that is leaky at higher frequencies, and a simultaneous propagation effect, which is rather general and which occurs when the line's relative cross-sectional dimensions are changed. The final section of the paper is concerned with three important recent developments: (a) the new effects that arise when the frequency is raised still higher and leakage occurs into an additional surface wave, (b) a basic and unexpected discovery relating to improper real modes, which are nonphysical but which can strongly influence the total physical field under the right circumstances, and (c) the important practical issue of how leakage behavior is modified when the circuit is placed into a package.