Let me start by telling you a story of an 18th-century french mathematician and physicist, Joseph Fourier. When Fourier was constructing his house, he ran into an serious problem: How do you build a wine cellar that always kept the right temperature? This is a rather important problem to solve for most Europeans. To describe the problem more mathematically: Fourier wanted to find solutions of the heat equation subject to particular boundary conditions (b.c.). With the right modelling of the b.c.'s, it is possible to construct exact solutions of the heat equation. This was one (among many) of the motivations for Joseph Fourier to develop, what we now call, Fourier series and which is not only an important tool for mathematical subjects but also used in signal processing, control systems, electrical engineering, mechanical engineering, and many more. For example, Fourier series allowed us to develop signal processing which gave us long distance communications a long time ago and the ability to use digital circuits in more recent times. Note that this example does not fall into the standard definition of an integrable models, but rather into the definition of exactly solvable models.

The subjects of what could be considered integrable systems goes far back. These subjects are the dynamical systems that can be solved by integrating them, but the integrals could not always be expressed explicitly in closed form. Famous examples include the Kepler problem which describes the motions of planets.

The first example of what we now would define as an integrable system was found by John Russel in 1834, who observed the first `solitary wave´ in an canal. This would eventually lead to the discovery and study of solitons (whose name derives from the solitary waves) as solutions of non-linear wave equations.

It is possible to recreate the waves that Russel found. Below, you can find two spectacular videos by Laboratoire Interdisciplinaire CARNOT de Bourgogne of these solitions in action:

If you are studying natural sciences or engineering, then you will eventually come across the theory of special functions. Examples of these include the Euler Gamma function, the famous Riemann zeta function, Airy functions, Bessel functions, elliptic functions, and classical orthogonal polynomials.

Special functions come in various forms, from simple expressions in terms of sums or integrals to horrifically complex-looking formulae, as well as all the intricate ways they can be related to each other.

In ages past, all these information on special functions were collected in huge books of tables, which were used for all kinds of purposes outside of mathematics; from navigation to construction. Today, all that have been collected in the National Institute of Standards and Technology Digital Library of Mathematical Functions. 

With the discovery of quantum mechanics, it became clear that physics at atomic scales were governed by partial differential equations, or commonly referred to as eigenvalue equations, obtained by a Hamiltonian operator.
The solutions of these equations are often found in terms of special functions, which helped us gain an understanding of the behavior of quantum particles. Our understanding of the atomic world has lead to the technological advancements that we use. For example, if we did know the exact behavior of electrons it would be impossible to construct the transistors needed for the circuits found in electronics.

Today, we are dealing with understanding so-called strongly correlated systems. These are systems where the strong interactions between particles lead to novel phenomena, such as (type-II) superconductivity and topological insulators. The strong interaction also makes traditional analytical methods and computer calculations ineffective. If we obtain an exact understanding of these phenomena, then it will be possible to solve important issues such as the electricity shortages by having ultra-efficient grids, revolutionize transportation with e.g. maglev trains, build new technologies such as the quantum computer, and many more.