In this thesis we present studies of scattering amplitudes on the celestial sphere at null infinity (celestial amplitudes), the cluster adjacency structure of scattering amplitudes in planar maximally supersymmetric Yang-Mills theory (N=4 SYM), and a method to derive symbol letters for loop amplitudes in N=4 SYM. First, we show that n-particle celestial gluon tree amplitudes take the form of Aomoto-Gelfand hypergeometric functions and Gelfand A-hypergeometric functions. We then study conformal properties, conformal partial wave decomposition, and the optical theorem of four-particle celestial amplitudes in massless scalar phi-cubed theory and Yang-Mills theory. Subsequently, we derive single- and multi-soft theorems for celestial amplitudes in Yang-Mills theory. Second, we provide computational evidence that each rational Yangian invariant in N=4 SYM has poles that are cluster adjacent (belong to the same cluster in the Gr(4,n) cluster algebra) through the Sklyanin bracket test. We also use this bracket test to study cluster adjacency of the symbol of one-loop NMHV amplitudes in N=4 SYM. Finally, we suggest an algorithm for computing symbol alphabets from plabic graphs by solving matrix equations of the form C.Z = 0 to associate functions on Gr(m,n) to parameterizations of certain cells in Gr(k,n) indexed by plabic graphs. For m=4 and n=8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-particle amplitude from four plabic graphs.
