In addition to several interesting facts, new curiosities, beautiful patterns, curious numbers and equations, Bhaskara pairs, palindromic and pandigital squares, exclusionary, biperiod, and tridigital squares are discussed in this chapter. Shortcuts in computing and methods of fast detection of perfect squares based on the quadratic residue test are explained. To further stimulate interest, two puzzles based on squares are dealt with. The power of the sum of digits, cubes containing their cube roots, perfect powers having like digits, squares with curious reversal properties, and automorphic numbers are also discussed.