The thesis studies and solves the Apollonius’ problem, which is: Three objects are given, a circle, a line, or a point. Construct a circle that touches all three given objects. The problem is subdivided into ten constructive problems. All ten problems, some basic definitions, theorems and simpler construction problems needed for the thesis are listed in the introduction. The thesis consists of three chapters. The first chapter describes the constructions of Apollonius’ problems in which there are no circles, or in other words, in which all given elements are either points or lines. The chapter is divided into four subchapters - each containing a solution to one constructive problem. The second chapter describes the constructions of Apollonius’ problems in which one or two given elements are circles and others are points and/or lines. This chapter is divided into five subchapters with each containing a solution to one constructive problem. The third chapter describes the construction of the original Apollonius’ problem, in which all three given elements are circles. All chapters study the methods of solving a constructive problem and stages of solving one (analysis, construction, proof, and discussion). Also, some problems are solved in more than one way. The stages of solving the constructive problems are described in detail. Some problems are divided into several cases depending on the positions of given objects and are solved separately as constructive problems. The problems are arranged by complexity and are substantiated with a list of images constructed using GeoGebra.