Mathematical physics is the scientific discipline concerned with "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories."
It can be seen as underpinning both theoretical physics and computational physics.

Scope of the subject
The great 17th century mathematician and physicist Isaac Newton developed a wealth of new mathematics, in an informal way, to solve problems in physics, including calculus and several numerical methods (most notably Newton's method). James Clerk Maxwell, Lord Kelvin,George Gabriel Stokes, William Rowan Hamilton, and J. Willard Gibbs were mathematical physicists who had a profound impact on 19th century science. Revolutionary mathematical physicists at the turn of the 20th century included the mathematician David Hilbert who devised the theory of Hilbert spaces for integral equations which would find a major application in quantum mechanics. Paul Dirac used algebraic constructions to produce a relativistic model for the electron, predicting its magnetic moment and the existence of its antiparticle, the positron. Albert Einstein's special relativity replaced the Galilean transformations of space and time with Lorentz transformations, and his general relativity replaced the flat geometry of the large scale universe by that of a Riemannian manifold, whose curvature replaced Newton's gravitational force. Other prominent mathematical physicists include Carl Friedrich Gauss, Jules-Henri Poincaré, Richard Feynman, Roger Penrose, and Satyendra Nath Bose. Carl Friedrich Gauss is largely considered to be one of the three greatest mathematicians of all time. His influence in mathematical physics is largely felt by his developing of the mathematical field non-Euclidean Geometry, which Albert Einstein's General Theory of Relativity as well as our understanding of the event horizon in black holes rely so heavily on.

Prominent mathematical physicists
The term 'mathematical' physics is also sometimes used in a special sense, to distinguish research aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. Mathematical physics in this sense covers a very broad area of topics with the common feature that they blend pure mathematics and physics. Although related to theoretical physics, 'mathematical' physics in this sense emphasizes the mathematical rigour of the same type as found in mathematics. On the other hand, theoretical physics emphasizes the links to observations and experimental physics which often requires theoretical physicists (and mathematical physicists in the more general sense) to use heuristic, intuitive, and approximate arguments. Such arguments are not considered rigorous by mathematicians. Arguably, rigorous mathematical physics is closer to mathematics, and theoretical physics is closer to physics.
Such mathematical physicists primarily expand and elucidate physical theories. Because of the required rigor, these researchers often deal with questions that theoretical physicists have considered to already be solved. However, they can sometimes show (but neither commonly nor easily) that the previous solution was incorrect.
The field has concentrated in three main areas: (1) quantum field theory, especially the precise construction of models; (2) statistical mechanics, especially the theory of phase transitions; and (3) nonrelativistic quantum mechanics (Schrödinger operators), including the connections to atomic and molecular physics.
The effort to put physical theories on a mathematically rigorous footing has inspired many mathematical developments. For example, the development of quantum mechanics and some aspects of functional analysis parallel each other in many ways. The mathematical study of quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous quantum field theory has brought about progress in fields such as representation theory. Use of geometry and topology plays an important role in string theory. The above are just a few examples. An examination of the current research literature would undoubtedly give other such instances.

Mathematically rigorous physics

Notes

The Classics

Sir Harold Jeffreys and Bertha Swirles (Lady Jeffreys), Methods of Mathematical Physics, third revised edition (Cambridge University Press, 1956 — reprinted 1999). ISBN 0-521-66402-0, ISBN 978-0-521-66402-8.
Eugene Butkov, Mathematical Physics. Addison Wesley, 1968.
Ivar Stakgold, Boundary Value Problems of Mathematical Physics. Vols. I and II. Macmillan, 1970.
Mary L. Boas, Mathematical Methods in the Physical Sciences. John Wiley & Sons, 3 ed., 2005.
George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists. Academic Press, 1995.
Jon Mathews and R.L. Walker, Mathematical Methods of Physics, 2e, Addison-Wesley, 1970. ISBN 0-8053-7002-1