The main aim of this thesis is to study certain extensions of Fourier transforms and Bessel potentials. The transforms can be written as composites (or compositions) of some elementary transformations, Fourier operators and Bessel potentials. The Bessel potentials have first been introduced by Aronszajn and Smith. Further, the extended Bessel potentials which are weighted forms of the operators previously considered, can be expressed as composites of the extended Fourier transforms.