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Fifteen years prior to their discovery in 1979, Refsdal (1964) outlined how gravitationally lensed quasars might be used to determine the Hubble constant. Astronomers have spent the quarter century since their discovery working out the difficult details not considered in Refsdal's seminal papers.

The difficulties encountered fall into two broad categories - measurement and modeling. Time delays can be hard to measure if the fluxes of the images do not vary, or if the images are faint, or if they lie very close to each other. Modeling gravitational potentials with a small number of constraints is likewise difficult, either because the lens geometry is complex or because the data poorly constrain the most important aspects of the gravitational potential. We will argue that these difficulties are surmountable, both in principle and in practice, and that an effort considerably smaller than that of the HST Hubble Constant Key Project will yield a considerably smaller uncertainty in the Hubble constant, H0.

While the number of systems with measured time delays is small, their interpretation implies a value for H0, which, given our current understanding of the dark matter distributions of galaxies, is formally inconsistent with that obtained using Cepheids. The Key Project value of H0 = 72 ± 8 km s-1 Mpc-1 (Freedman et al. 2001) is consistent with the lens data only if the lens galaxies have significantly less dark matter than is expected theoretically or has been measured for other early-type galaxies. While it is premature to argue for replacing the local estimates, we hope to persuade the astronomical community that the time delay result deserves both careful attention and further study.

Interpreting time delays requires a model for the gravitational potential of the lens, and in most cases the uncertainties in the model dominate the uncertainty in H0. Thus, the main focus of this review will be to explain the dependence of time delays on gravitational potentials. We start in Section 2 by introducing the time delay method and illustrating the physics of time delays with a series of simple models. In Section 3 we review a general mathematical theory of time delays to show that, for most lenses, the only important parameter of the model is the mean surface density of the lens at the radius of the images. In Section 4 we discuss the effects of the environment of the lens on time delays. We review the data on the time delay lenses in Section 5 and their implications for the Hubble constant and dark matter in early-type galaxies in Section 6. The present time delay lenses have a degeneracy between H0 and the amount of dark matter, so in Section 7 we outline several approaches that can eliminate the degeneracy. Finally, in Section 8 we discuss the future of time delays. Unless otherwise stated, we assume a flat, Omegam = 0.3, OmegaLambda = 0.7 cosmological model.

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