Molecular biology is becoming a computationally intense realm of contemporary science and faces some of the current grand scientific challenges. In its context, tools that identify, store, compare and analyze effectively large and growing numbers of bio-sequences are found of increasingly crucial importance. Biosequences are routinely compared or aligned, in a variety of ways, to infer common ancestry, to detect functional equivalence, or simply while searching for similar entries in a database. A considerable body of knowledge has accumulated on sequence alignment during the past few decades. Without pretending to be exhaustive, this paper attempts a survey of some criteria of wide use in sequence alignment and comparison problems, and of the corresponding solutions. The paper is based on presentations and literature given at the Workshop on Sequence Alignment held at Princeton, N.J., in November 1994, as part of the DIMACS Special Year on Mathematical Support for Molecular Biology.