A mechanism was developed explaining why bubbles can enter a capillary of diameter smaller than the diameter of the bubble. A cocurrent flow of liquid creates a hydrodynamic pressure field that deforms the bubble surface and promotes the formation of a saddle-like interface near the capillary mouth. A mechanistic model for the bubble to deform is given. Even small bubbles would not enter the capillary if there is no flow of liquid. Bubbles of a diameter larger than the diameter of a capillary - or the side of a capillary of square cross section - can be forced to enter the capillary only if there is a cocurrent flow of liquid. A high-speed video imaging system was used to record interesting aspects of this phenomenon. On the average, the entire process for the bubble to enter the capillary lasts less than 200 ms. This problem has interesting applications in two-phase flows inside ceramic monoliths, the flow of foams inside porous media and in the flow of blood into capillary vessels.