Amyloid fibrils are a polymeric aggregate of protein. The fibrils are typically on the order of micrometers long, with widths of 10-20 nm. They are generally regarded as stiff, and nonbranching. It is well-known that similar synthetic polymers and biopolymers such as DNA and polysaccharides, have a tendency to form liquid crystalline phases when incubated under appropriate conditions. Here we show that amyloid fibrils from the protein hen lysozyme can similarly form liquid crystal phases. The most common phase observed is the nematic. Alignment can persist for several centimeters. When the fibrils are freeze-thawed to shorten them, similar phases form but at higher concentrations, confirming the importance of the aspect ratio of the fibrils. Freeze-thawed fibrils are also seen to form "tactoids", discrete liquid crystalline structures. The addition of NaCl to the solutions appears to only have a minor effect, while the effect of pH appears much more significant. We propose that the consideration of amyloid fibrils as polymer analogues should open new routes to explore in the burgeoning field of biomaterials.