The amplitudes of fast radio bursts (FRBs) can be strongly modulated by plasma lenses in their host galaxies, including that of the repeating FRB 121102 at ∼1 Gpc luminosity distance. Caustics require the lens' dispersion measure depth ( ), scale size (a), and distance from the source ( ) to satisfy . Caustics produce strong magnifications ( ) on short timescales ( hours to days) that appear as narrow spectral peaks (0.1-1 GHz). They also suppress the flux density in longer-duration (∼months) troughs. Multiply imaged bursts will arrive differentially by to tens of ms with different apparent dispersion measures, pc cm−3. When differing by less than the burst width, interference effects in dynamic spectra will be seen. Larger arrival time perturbations may mask any underlying periodicity with period . Strong lensing requires sources smaller than , which includes compact objects such as neutron star magnetospheres but excludes active galactic nuclei. We discuss constraints on densities, magnetic fields, and locations of plasma lenses related to the conditions needed for lensing to occur. Much of the phenomenology of the repeating FRB source FRB 121102 can be accounted for in this picture, which can be tested by obtaining wideband spectra of bursts (from to 10 GHz and possibly higher) that will also help characterize the plasma environment near FRB sources. A rich variety of phenomena is expected from an ensemble of lenses near an FRB source.