We propose that turbulent mixing layers are common in the interstellar medium (ISM) of our Galaxy and selected external galaxies, with many layers per kiloparsec along typical lines of sight. Initiated by shear flows at the boundaries of hot and cold gas, these mixing layers produce intermediate-temperature gas at TBAR almost-equal-to 10(5.0)-10(5.5) K that radiates strongly in the optical, ultraviolet, and extreme ultraviolet. Expanding on the idea of Begelman & Fabian, we have modeled these layers under the assumptions of rapid mixing and mean steady flow. By including the effects of nonequilibrium ionization and self-photoionization of the gas as it cools after mixing, we predict the intensities of numerous optical, infrared, and ultraviolet emission lines, as well as absorption column densities of C IV, N V, Si IV, and O VI. Mixing layers can produce the line ratios [S II]/Halpha almost-equal-to 0.5, [N II]/Halpha almost-equal-to 0.3-0.7, and [O II]/[O III] almost-equal-to 2-3, in good agreement with observations of diffuse ionized gas and ''interstellar froth'' in Magellanic irregular galaxies and some edge-on spiral galaxies, and with the extended component of diffuse ionized gas in the Seyfert galaxy NGC 1068. Mixing layers also provide a transfer mechanism from hot to cold gas for reradiation of the energy deposited in the hot interstellar gas by supernovae and stellar winds. The line intensities from a mixing layer are proportional to the gas pressure. In the Galaxy, at a pressure P/k(B) almost-equal-to 3000 cm-3 K, the layers can account for (1) between 2% and 10% of the diffuse Halpha in the Galactic disk; (2) a significant fraction (possibly >30%) of the diffuse Halpha at high latitude, with correct line ratios of [S II], [N II], [O II], and [O III]; (3) all of the high-latitude C IV lambda1550 emission observed by Martin & Bowyer with the observed ratios of O III] lambda1663 to C IV lambda1549 emission and C IV emission to absorption; and (4) the observed column density ratios of the highly ionized species C IV, N V, O VI, and Si IV if one posits some grain depletion. Some approximately 20% of the supernova power and 75 M. yr-1 may be processed through mixing layers. Thus, these processes could have dramatic effects on the energy budget of the Galactic fountain. Further observations of the emission lines O III] lambda1663, [O III] lambda5007, and O VI lambdalambda1032, 1038 at high latitude are important to test our models and to understand the nature of ''frothy'' diffuse ionized gas in the ISM.