This thesis defines the Biot-Savart operator (BS) and describes several of its properties. The definition of the Biot-Savart operator requires basic understanding of the laws of magnetostatics, as well as the Biot-Savart law, whose application is in electrodynamics course. Furthermore, the Hodge decomposition theorem is an integral part of understanding the Biot-Savart operator, therefore it has been defined, along with the examples of vector fields which are discussed in that theorem. Amongst other things, in this thesis we show that the Biot-Savart operator is an adjoint, bounded operator, and we describe its image and how one can find the kernel of this operator. The last chapter presents a short historical overview, as well as a description of a widerange application of the Biot-Savart operator in plasma physics, magnetohydrodynamics, molecular biology and other branches of natural sciences. We also define several variables, such as helicity and writhing number, along with their connection with the Biot-Savart operator and a description of their upper bounds. Finally, at the end of the thesis, we present in detail the DNA supercoiling phenomenon, with a description of its three qualities: linking number, writhing number and twist, in addition to the description of their mutual relationship