in this talk, we will first briefly review the generic chaining method and dudley's inequality. then, we will show the application of chaining methods in random dimensionality reduction. we will mainly consider dimensionality reduction with johnson-lindenstrauss type embeddings, which are random matrix constructions to reduce the dimension while approximately preserving euclidean inter-point distances in the data set. in particular, we will consider ``fast'' and ``sparse'' johnson-lindenstrauss embeddings. finally, we summarize some applications of chaining methods in data streaming algorithms and dictionary learning. the material of this talk is based on some lecture notes by jelani nelson and some research work by sjoerd dirksen, jean bourgain and jelani nelson.