One of the major challenges in denotational semantics is the construction of a fully abstract semantics for a higher-order sequential programming language. For the past fifteen years, research on this problem has focused on developing a semantics forPCF, an idealized functional programming language based on the typed lambda calculus. Unlike most practical languages, PCF has no facilities for observing and exploiting the evaluation order of arguments to procedures. Since we believe that these facilities play a crucial role in sequential computation, this paper focuses on a sequential extension of PCF, called SPCF, that includes two classes of control operators: a possibly empty set of error generators and a collection of catch and throw constructs. For each set of error generators, the paper presents a fully abstract semantics for SPCF. If the set of error generators is empty, the semantics interprets all procedures—including catch and throw—as Berry-Curien sequential algorithms. If the language contains error generators, procedures denote {\it manifestly sequential} functions. The manifestly sequential functions form aScott domain that is isomorphic to a domain of decision trees, which is the natural extension of the Berry-Curien domain of sequential algorithms in the presence of errors.