$t$-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on $t$-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on $t$-cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of $t$-cores in number theoretic aspects of partition theory. This includes the modularity of $t$-core partition generating functions, the existence of $t$-core partitions, asymptotic formulas and arithmetic properties of $t$-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of $t$-core partitions, which include relations between core partitions and self-conjugate core partitions, a $t$-core crank explaining Ramanujan's partition congruences, and relations with class numbers.