In this paper, firstly, we apply the shooting method together with the Bolzano’s theorem and the maximum principle to present the existence, uniqueness, and qualitative properties of solutions to nonlinear third-order two-point boundary value problems on the half-line. And then, by employing the matching method, the existence, uniqueness, and qualitative properties of solutions to nonlinear third-order three-point boundary value problems on the whole real line are obtained. Finally, as applications, we mainly present the existence, uniqueness, and qualitative properties of solutions of the Blasius equations $y’‘’+yy’‘=0$ with one of the following boundary conditions $$ y’(-\infty)=C,~ y(0)=0,~ y’(+\infty)=1~ \hbox{with}~ 0\leq C<1, $$ $$ y’(-\infty)=C,~ y’‘’(0)=0,~ y’(+\infty)=1~ \hbox{with}~0\leq C<1, $$ which arise from the laminar mixing layer between two parallel flows with different velocities.