Integrability of differential equations with fluid mechanics application: from Painleve property to the method of simplest equation

We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order has the Painleve property if the only movable singularities connected to this equation are single poles. The importance of this property can be seen from the Ablowitz-Ramani-Segur conhecture that states that a nonlinear PDE is solvable by inverse scattering transformation only if each nonlinear ODE obtained by exact reduction of this PDE has the Painleve property. The Painleve property motivated motivated much research on obtaining exact solutions on nonlinear PDEs and leaded in particular to the method of simplest equation. A version of this method called modified method of simplest equation is discussed below.