The discrete Chebyshev polynomials tn(x,N) are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the points x = 0, 1, · · · ,N-1, N being a fixed positive integer. In this thesis, by using a double integral representation, we derive asymptotic expansions for tn(aN,N + 1) in the double scaling limit, namely, N→∞and n/N→b, where b ∈ (0, 1) and a ∈ (-∞,∞). Our discussion is divided into two parts. In the first part, we obtain two expansions for tn(aN,N+1) when the parameter b is a fixed constant in the interval (0, 1). One expansion involves the confluent hypergeometric function and holds uniformly for a ∈ [0, 1/2 ], and the other involves the Gamma function and holds uniformly for a ∈ (-∞, 0). Both intervals of validity of these two expansions can be extended slightly to include a neighbourhood of the origin. Asymptotic expansions for a ≥ 1/2 can be obtained via a symmetry relation of tn(aN,N +1) with respect to a = 1/2 . Asymptotic formulas for small and large zeros of tn(x,N + 1) are also given. In the second part, we continue to investigate the behaviour of these polynomials when the parameter b approaches the endpoints of the interval (0, 1). While the case b→1 is relatively simple (since it is very much like the case when b is fixed), the case b→0 is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities n, x and xN/n2, and different special functions have been used as approximants, including Airy, Bessel and Kummer functions.