This manual is written on the basis of courses of lectures delivered for students of the Faculty of Mathematics of VSU in 2001-2003 . In the first lecture of the course, the main properties of the Laplace transformation are studied. The second lecture calculates Laplace's transformations of basic elementary functions and introduces the inverse Laplace transform. In the third lecture, on the basis of the first and second decomposition theorems, the inverse Laplace transform is calculated for a number of functions. Examples of solving problems for partial-derivative equations by direct and inverse Laplace transformations are given in the fourth lecture of the course. Finally, the final section of the manual contains examples of two tasks that are usually offered to students who have attended the given course to control the assimilation of material. Here are the solutions of the proposed tasks with the corresponding explanations .. When presenting material and solving tasks, the Mathematica character program package is widely used. Comments on the Mathematica package commands used can be found in the reference publications indicated at the end of the manual in the list of recommended references