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Lutetium is a chemical element; it has symbol Lu and atomic number 71. It is a silvery white metal, which resists corrosion in dry air, but not in moist air. Lutetium is the last element in the lanthanide series, and it is traditionally counted among the rare earth elements; it can also be classified as the first element of the 6th period transition metals. Lutetium was independently discovered in 1907 by French scientist Georges Urbain, Austrian mineralogist Baron Carl Auer von Welsbach, and American chemist Charles James. All of these researchers found lutetium as an impurity in the mineral ytterbia, which was previously thought to consist entirely of ytterbium. The dispute on the priority of the discovery occurred shortly after, with Urbain and Welsbach accusing each other of publishing results influenced by the published research of the other; the naming honor went to Urbain, as he had published his results earlier. He chose the name lutecium for the new element, but in 1949 the spelling was changed to lutetium. In 1909, the priority was finally granted to Urbain and his names were adopted as official ones; however, the name cassiopeium (or later cassiopium) for element 71 proposed by Welsbach was used by many German scientists until the 1950s. Lutetium is not a particularly abundant element, although it is significantly more common than silver in the Earth's crust. It has few specific uses. Lutetium 176 is a relatively abundant (2.5%) radioactive isotope with a half life of about 38 billion years, used to determine the age of minerals and meteorites. Lutetium usually occurs in association with the element yttrium and is sometimes used in metal alloys and as a catalyst in various chemical reactions. 177Lu DOTA TATE is used for radionuclide therapy (see Nuclear medicine) on neuroendocrine tumours. Lutetium has the highest Brinell hardness of any lanthanide, at 890–1300 MPa.

A lutetium atom has 71 electrons, arranged in the configuration [Xe] 4f145d16s2. Lutetium is generally encountered in the 3+ oxidation state, having lost its two outermost 6s and the single 5d electron. The lutetium atom is the smallest among the lanthanide atoms, due to the lanthanide contraction, and as a result lutetium has the highest density, melting point, and hardness of the lanthanides. As lutetium's 4f orbitals are highly stabilized only the 5d and 6s orbitals are involved in chemical reactions and bonding; thus it is characterized as a d block rather than an f block element, and on this basis some consider it not to be a lanthanide at all, but a transition metal like its lighter congeners scandium and yttrium.

Lutetium is usually found in the +3 oxidation state, like most other lanthanides. However, it can also be in the 0, +1 and +2 states as well.

Lutetium occurs on the Earth in form of two isotopes: lutetium 175 and lutetium 176. Out of these two, only the former is stable, making the element monoisotopic. The latter one, lutetium 176, decays via beta decay with a half life of 3.78; it makes up about 2.5% of natural lutetium. To date, 40 synthetic radioisotopes of the element have been characterized, ranging in mass number from 149 to 190; the most stable such isotopes are lutetium 174 with a half life of 3.31 years, and lutetium 173 with a half life of 1.37 years. All of the remaining radioactive isotopes have half lives that are less than 9 days, and the majority of these have half lives that are less than half an hour. Isotopes lighter than the stable lutetium 175 decay via electron capture (to produce isotopes of ytterbium), with some alpha and positron emission; the heavier isotopes decay primarily via beta decay, producing hafnium isotopes. The element also has 43 known nuclear isomers, with masses of 150, 151, 153–162, and 166–180 (not every mass number corresponds to only one isomer). The most stable of them are lutetium 177m, with a half life of 160.4 days, and lutetium 174m, with a half life of 142 days; these are longer than the half lives of the ground states of all radioactive lutetium isotopes except lutetium 173, 174, and 176.

Because of production difficulty and high price, lutetium has very few commercial uses, especially since it is rarer than most of the other lanthanides but is chemically not very different. However, stable lutetium can be used as catalysts in petroleum cracking in refineries and can also be used in alkylation, hydrogenation, and polymerization applications. A nitrogen doped lutetium hydride may have a role in creating room temperature superconductors at 10 kbar. Lutetium aluminium garnet (|Al5Lu3O12) has been proposed for use as a lens material in high refractive index immersion lithography. Additionally, a tiny amount of lutetium is added as a dopant to gadolinium gallium garnet, which is used in magnetic bubble memory devices. Cerium doped lutetium oxyorthosilicate is currently the preferred compound for detectors in positron emission tomography (PET). Lutetium aluminium garnet (LuAG) is used as a phosphor in light emitting diode light bulbs. Aside from stable lutetium, its radioactive isotopes have several specific uses. The suitable half life and decay mode made lutetium 176 used as a pure beta emitter, using lutetium which has been exposed to neutron activation, and in lutetium–hafnium dating to date meteorites. The synthetic isotope lutetium 177 bound to octreotate (a somatostatin analogue), is used experimentally in targeted radionuclide therapy for neuroendocrine tumors. Indeed, lutetium 177 is seeing increased usage as a radionuclide in neuroendocrine tumor therapy and bone pain palliation. Research indicates that lutetium ion atomic clocks could provide greater accuracy than any existing atomic clock. Lutetium tantalate (LuTaO4) is the densest known stable white material (density 9.81 g/cm3) and therefore is an ideal host for X ray phosphors. The only denser white material is thorium dioxide, with density of 10 g/cm3, but the thorium it contains is radioactive. Lutetium is also a compound of several scintillating materials, which convert X rays to visible light. It is part of LYSO, LuAg and lutetium iodide scintillators. Lutetium (177Lu) vipivotide tetraxetan is a therapy for prostate cancer, FDA approved in 2022.

Like other rare earth metals, lutetium is regarded as having a low degree of toxicity, but its compounds should be handled with care nonetheless: for example, lutetium fluoride inhalation is dangerous and the compound irritates skin. Lutetium nitrate may be dangerous as it may explode and burn once heated. Lutetium oxide powder is toxic as well if inhaled or ingested. Similarly to the other rare earth metals, lutetium has no known biological role, but it is found even in humans, concentrating in bones, and to a lesser extent in the liver and kidneys. Lutetium salts are known to occur together with other lanthanide salts in nature; the element is the least abundant in the human body of all lanthanides. Human diets have not been monitored for lutetium content, so it is not known how much the average human takes in, but estimations show the amount is only about several micrograms per year, all coming from tiny amounts absorbed by plants. Soluble lutetium salts are mildly toxic, but insoluble ones are not.