Razor having a hair-cutting fiber having a hair-cutting section attached to a support in such a position relative to a skin engaging surface that the hair-cutting section is exposed to a user's skin for cutting through hairs when the skin engaging surface is in contact with the skin and moved over the skin. The hair-cutting section has a fiber diameter smaller than or equal to 7 μm and is made from a fiber material having a tensile strength larger than or equal to 5 GPa. A shaving method using such a razor is also described.