Before we left our own planet the first time, we had very little information on other planets. There was much room for fantasy, and some ideas have proven unrealistic. One of them is obviously the theory of ice and air filled caves on the Moon, postulated in science fiction stories and shown in 1950s science fiction B-movies.. But still the last word on caves on Moon is not spoken.
The moon has no atmosphere and so, despite the enormous temperature changes by day and night, no weathering happens. Lava tubes are formed by flowing lava, where the outside cools down and forms a tunnel, while the lava still flows inside. This could happen on Moon, if there is volcanism. Theoretically.
The volcanism on moon is proven by samples, brought back by various Apollo missions. This volcanism happened about 3.5 billion years ago, which was found out using modern geophysical dating methods on the rock samples. The Moon volcanism could have produced numerous lava tubes. Such lava tubes on earth are destroyed within 10,000 years by weathering. But the Moon has no atmosphere and thus no weathering, so they could have lasted until today. Just the bit about air and ice is extremely unlikely. Moon most likely never had any atmosphere, and if the caves contained gas as a remains of the volcanic processes (which obviously would be unsuitable to breathe) this gas would have left long ago, as normal rocks get cracks and are definitely not air tight.
Like the caves on Mars, the caves on Moon would be very valuable. They would provide shelter against extreme temperature changes, solar radiation, and meteors. And some people believe, they could contain messages from alien visitors which were on Earth long time ago and left it there for us to discover, when we are "grown up"...
So far no caves have been detected on the moon, but there there is one possible cave entrance, actually a vertical shaft in the surface, which could be the entrance to a lava tube, a place where the ceiling collapsed. The discovery is of great importance to future moon missions, so the search for more possible cave entrances and caves will be continued.