It is well known that solutions of one-dimensional Schrödinger equation in free space in the form of coherent states, i.e., wave packets which are Gaussian functions of the coordinate, \[ \psi(x,t)=A(t)\exp[-(a(t)/2)x^2], \] can be found in an exact analytical form, a part of which is provided by an exact solution to the equation of the Riccati type for complex function \(a(t)\). This fact gives rise to various links between the linear one-dimensional Schrödinger equation and nonlinear ODEs of the Riccati type. Such links are a general topic of the present book. In particular, links of the Schrödinger equation to more specific nonlinear ODEs, such as the Bernoulli equation and Ermakov equation, are included too. Further, several chapters of the book address solutions of dissipative extensions of the Schrödinger equation, chiefly in the form of Gaussians. A non-orthodox generalization of quantum mechanics, in the form of a nonlinear dissipative Schrödinger equation, is considered too. A separate chapter addresses Riccati equations which naturally occur in other physically relevant settings, such as nonlinear optics and the dynamics of matter waves in Bose-Einstein condensates, as well as soliton-generating PDEs – in particular, the Burgers and Korteweg-de Vries equations.