The numbers in the Ulam sequence, first defined by S. Ulam, have been of considerable interest to number theorists because of their unusual and mysterious properties. Based on the computations of the first 2.1 billion Ulam numbers and non-Ulam Numbers (unulam numbers and superulam numbers), several interesting observations are discussed. The distribution of gaps between Ulam numbers is also dealt with in this chapter. Some gap sizes are found to occur frequently, whereas some of the gap sizes are not found to occur, such as 6, 11, 14, 16, 18, 21, 23, 26, 28, 31, 33, 36, 38, 40, 43, 45, 48, and 50. The possible reasons for frequently occurring gaps, missing gaps, and infrequently occurring gaps are discussed in the chapter, which can be further explored.