The Korteweg - de Vries equation traces back to the work of Boussinesq in the 1870's. Boussinesq in turn was motivated in part to explain the solitary waves observed first by Russell in the 1830's. Boussinesq's work, and the later, related work of Korteweg and de Vries, went largely unremarked by the fluid mechanics and the mathematics communities until the second half of the $20^{th}$ century. Since that time, there has been an explosion of activity centered around the Korteweg-de Vries equation (KdV equation) and its relatives, and the subject of nonlinear, dispersive waves has become a major preoccupation of mathematicians.

After some brief historical remarks, various classes of nonlinear dispersive equations are introduced. The focus of the talk is solitary-wave solutions of nonlinear dispersive equations. More specifically, when they cannot be found explicitly, we need to know whether they exist and, if so, what properties they possess. To answer such question, various techniques are useful and I will outline some of the work in this area. As time allows, I would continue the discussion with comments about recent stability results for explicit solitary-wave solutions of systems of KdV equations coupled through nonlinearity,
$$u_t+u_{xxx}+P(u, v)_x=0,$$
$$v_t +v_{xxx}+Q(u,v)_x=0,$$
where $u=u(x,t), v=v(x,t)$ are functions defined for $x\in (-\infty, \infty)$ and $t\in [0, \infty)$ and $P$ and $Q$ are quadratic polynomials of the form
\[P(u,v)=Au^2+Buv+Cv^2\] and \[Q(u,v)=Du^2+Euv+Fv^2\] with $A, B, \cdots, F$ specified real constants.