The Shuffle Theorem of Carlsson and Mellit gives a well-studied combinatorial expression for the bigraded Frobenius characteristic of the ring of diagonal harmonics. The Rational Shuffle Theorem of Mellit and the Delta Conjecture proposed by Haglund, Remmel and Wilson are two natural generalizations of the Shuffle Theorem. The Primary goal of this dissertation is to prove some special cases of the conjectures, and compute the Schur function expansion of the corresponding symmetric function expressions.