Optical coherence tomography (OCT) provides an alternative to physical sectioning that allows for imaging of living samples and even in vivo examination of cell structure and dynamics. There is, in the OCT community, a widely held belief that there exists a trade-off between transverse resolution and the thickness of the volume that may be imaged with a fixed focal plane. Efforts to overcome this trade-off have focused on the design optical elements and imaging hardware. I present a solution of the inverse scattering problem for optical coherence tomography (OCT) called interferometric synthetic aperture microscopy (ISAM) that provides resolution everywhere equal to the best resolution in the raw data (in the focal plane). I will discuss the connection between this solution, synthetic aperture radar, and the projection-slice theorem. I will give several examples of the method in use in biological systems and results from a recent clinical trial in breast cancer. Several import mathematical problems remain in this field. I will discuss a few of the more interesting and immediately useful examples.