Let X be the affine flag manifold of Lie type A_n-1⁽¹⁾ where n ≥ 3 and let W_aff be the associated affine Weyl group. The moment graph for X encodes the torus fixed points (which are elements of the affine Weyl group W_aff and the torus stable curves in X. Given a fixed point u ∈ W_aff and a degree d = (d₀, d₁, ..., d_n−1) ∈ ℤ_≥0ⁿ, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of X which can be reached from u′ ≤ u by a chain of curves of total degree ≤ d. In this thesis we give combinatorial formulas and algorithms for calculating these elements.