A literature review reveals many lines of evidence that both delayed rectifier and inward rectifier potassium channels are multi-ion pores. These include unidirectional flux ratios given by the 2--2.5 power of the electrochemical activity ratio, very steeply voltage-dependent block with monovalent blocking ions, relief of block by permeant ions added to the side opposite from the blocking ion, rectification depending on E--EK, and a minimum in the reversal potential or conductance as external K+ ions are replaced by an equivalent concentration of T1+ ions. We consider a channel with a linear sequence of energy barriers and binding sites. The channel can be occupied by more than one ion at a time, and ions hop in single file into vacant sites with rate constants that depend on barrier heights, membrane potential, and interionic repulsion. Such multi-ion models reproduce qualitatively the special flux properties of potassium channels when the barriers for hopping out of the pore are larger than for hopping between sites within the pore and when there is repulsion between ions. These conditions also produce multiple maxima in the conductance-ion activity relationship. In agreement with Armstrong's hypothesis (1969. J. Gen. Physiol. 54:553--575), inward rectification may be understood in terms of block by an internal blocking cation. Potassium channels must have at least three sites and often contain at least two ions at a time.