The one-dimensional Laplace transform method is widely used in engineering mathematics, where it has numerous applications. Particularly useful in problems where the mechanical or electrical driving force has discontinuities, for instance, acts for a short time only, [Kreyszig E., 1983]. This chapter consists of two sections: In section one, we give the definition and theorems of the one-dimensional Laplace transform for functions of only one independent variable. Also some properties of the one-dimensional Laplace transform are presented. In section two, we use the one-dimensional Laplace transform as a method to solve special types of the linear problems namely, linear ordinary differential equations (with or without delays) with constant coefficients and with or without initial conditions, linear one-dimensional Volterra integral and integro-differential equations and linear difference equations.