A Skolem sequence of order $t$ is a sequence $2t$ integers such that each integer between 1 and $t$ appears twice and two instances of the integer $k$ are $k$ apart. For example, 5242354311 is a Skolem sequence of order 5. These sequences, and their generalizations, are very interesting from a combinatorial point of view and have many applications. In this talk, we will discuss Skolem sequences and some generalizations: extended, Langford, and near-Skolem sequences. We will also discuss a few applications of these sequences, including integer partitioning and graph decompositions.