The linear least squares problem's sensitivities to perturbations are analyzed based on the first-order sensitivities of metric projections to perturbations of their sets. The latter is is a topic in real analysis and optimization theory that underlies the kind of perturbation theory used in numerical analysis. The general theory is applied to the linear least squares problem to obtain the following results. Simple formulas are derived that asymptotically equal the size of the minimal (in the Frobenius norm) backward errors. It is shown that such formulas can be used to evaluate condition numbers. For full rank problems, Frobenius norm condition numbers are determined exactly, and spectral condition numbers are determined within a factor of two. Necessary and sufficient criteria for well conditioning are established. A source of ill conditioning is found that explains the failure of simple iterative refinement for these problems. Some error bounds, including the one that LAPACK recommends, are found to unnecessarily overestimate the error. Finally, several open questions are described.