Radical, from Late Latin radicalis "of roots" and from Latin radix "root", may refer to:

in science and mathematics
A radical (chemistry) is an atom, molecule, or ion which is likely to take part in chemical reactions.
Radical symbol (√), a symbol used to indicate the square root or nth root
Radical of an algebraic group, a concept in algebraic group theory
Radical of an ideal, an important concept in abstract algebra
Radical of a ring, in ring theory, a branch of mathematics, a radical of a ring is an ideal of "bad" elements of the ring
Radical of a module, in the theory of modules, the radical of a module is a component in the theory of structure and classification
Jacobson radical, the Jacobson radical of a ring, R, consists of those elements in R which annihilate all simple right R-modules
Nilradical of a ring, the nilradical of a commutative ring is a nilpotent ideal, which is as large as possible
Nilradical of a Lie algebra, the nilradical of a lie algebra is a nilpotent ideal, which is as large as possible
Radical of an integer, a concept in number theory
Symmetric bilinear form, a concept in linear algebra
Radical of a Lie algebra, a concept in lie theory

in medicine
Radical surgery, surgery carried out in extreme circumstances and/or with extensive treatment

in linguistics
Root (linguistics), also called a "radical", the form of a word after any prefixes and suffixes are removed
Radical (Chinese character), part of a Chinese character under which it is indexed in a dictionary (also called bùshǒu 部首)
Radical, one of the three consonants in a Semitic root
Radical consonant, a consonant articulated with the root of the tongue

in politics
Political radicalism
Radicalism (historical), the Radical Movement which began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century
Radical left, another term for the far left
Radical right, another term for the far right
radical a person who shows a strong political stand point
Extremism, political standpoints that are perceived as revolutionary or extreme
Radical Whigs, influential early writers on Radicalism
Radical Reformation, an Anabaptist movement concurrent with the Protestant Reformation
Radical Republicans, a member of an influential group of American politicians during the Civil War reconstruction
Radical Party, one of a number of political parties around the world
Radicals (UK), parliamentary progressives who were part of the nineteenth-century Liberal coalition
Radical feminism, a current within feminism that focuses on patriarchy as a system of power
Radical veganism, a radical interpretation of veganism, usually combined with anarchism

in culture
Radical chic, a term coined by Tom Wolfe to describe the pretentious adoption of radical causes by celebrities, socialites, and high society

in architecture
Radical Baroque, an architectural style characterized by the curvature of walls and intersection of oval spaces

brands, titles and proper names
Radical Sportscars, an English sports car maker
Radical Entertainment, a video game developer based in Vancouver, Canada
Free Radical Design, a video game developer based in Nottingham, England 
Free Radicals (band), a jazz-funk musical group from Houston, Texas
Free Radical Centre, a centre for free radical chemistry and biotechnology that was established in 2005
Radical (Smack album), Smack's fourth studio album, released in 1988