The goal of this paper is to give a survey on the various computational and mathematical issues and progress related to the problem of providing efficient correctly-rounded elementary functions in floating-point arithmetic. We also aim at convincing the reader that a future standard for floating-point arithmetic should require the availability of a correctly-rounded version of a well-chosen core set of elementary functions. We discuss the interest and feasibility of this requirement. We also give answers to common objections we have received over the last 10 years.