The concept of partition is important in combinatorics, number theory and representation theory. It has attracted much attention in past centuries. In this thesis, the difference operators on functions of partitions and strict partitions are introduced to derive many new and classic hook-content formulas. As anapplication, several generalizations of Han-Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to hook lengths and contents are achieved. Many new hook-content formulas for self-conjugate and doubled distinct partitions are also derived. Motivated by number theory and modular representation theory of symmetric groups, t-core partitions are widely studied recently by many mathematicians. This thesis studies simultaneous core partitions and characterizes the largest size of (t; t+1; : : : ; t+p)-core partitions. We also verify Amdeberhan's conjectures on the number, the largest size and the average size of (t; t + 1)-core partitions with distinct parts.