This paper deals with a brief introduction to major remarkable discoveries of the "soliton" and the "inverse scattering transform" in the 1960s. The discovery of the soliton (or the solitary waves) began with the famous physical experiments of the Scottish Engineer and Naval Architect John Scott Russell in the Glasgow-Edinburgh Canal in 1834. The main objective of this paper is to introduce Scott Russell's vision to the reader. This was followed by the famous mathematical derivation of the Korteweg-de Vries (KdV) equation in 1895 for the propagation of the great solitary wave of Scott Russell in one direction on the free surface of water in a shallow canal. In 1965, Norman Zabusky and Martin Kruskal discovered the existence of solitons and the interaction of solitons from their computer experiments. In 1967, C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura developed an ingenious method for finding the exact solution of the KdV equation. This paper is also concerned with several fundamental nonlinear partial differential equations including the Korteweg and de Vries (KdV) equations, the nonlinear Schrodinger (NLS) equation, the Sine-Gordon (SG) equation and the Toda lattice equation. Thus, the soliton and the inverse scattering transform are now regarded as the major remarkable discoveries in mathematical sciences of the second half of the twentieth century. A brief comment is made on the discovery of compacton in the 1990s and physical interaction of two or more compactons. A compacton is a soliton with a finite compact support with "no" oscillatory trail. Interestingly, compactons can describe the intrinsic localized modes in anharmonic crystals. Concluding remarks are made on the recent developments of nonlinear wave phenomena and their applications to a wide variety of problems. (Contains 7 figures.)