Artificial left-handed (LH) materials have been realized recently using a two-dimensional array of repeated unit cells of conducting rods and split ring resonators. As a consequence, part of electromagnetic energy is stored in the structure, making the equivalent LH material lossy. In this Letter, the propagation of electromagnetic waves in lossy LH materials is investigated. By introducing a small loss, we easily show that the propagating constant in LH materials is negative, and furthermore, the index of refraction is negative. Then the reflection and transmission of a Gaussian beam in a lossy LH half space and a dielectric slab are investigated, where the lossy effect on propagating and evanescent waves has been intensively studied. For the case of LH half space, it is shown that the evanescent waves have been greatly amplified when the loss is small, and can be amplified to infinity when the LH material is lossless. This is a very interesting physical phenomenon where the lossless passive LH half space is an infinite amplifier, producing a physical singularity. In a lossless LH slab, however, the physical singularity is naturally cancelled, and hence such a slab can make a perfect image [Phys. Rev. Lett. 85 (2000) 3966]. For the case of lossy LH slab, a nearly-perfect image can still be obtained when the loss is small due to the amplification of evanescent waves. Numerical simulations are presented to support the above conclusions.