Advances in sequencing technologies and increased access to sequencing services have led to renewed interest in sequence and genome assembly. Concurrently, new applications for sequencing have emerged, including gene expression analysis, discovery of genomic variants and metagenomics, and each of these has different needs and challenges in terms of assembly. We survey the theoretical foundations that underlie modern assembly and highlight the options and practical trade-offs that need to be considered, focusing on how individual features address the needs of specific applications. We also review key software and the interplay between experimental design and efficacy of assembly.