  In this thesis we study the Durdevic theory of differential calculi on
quantum principal bundles within the domain of noncommutative geometry.
Throughout the exposition, an algebraic approach based on Hopf algebras is
employed. We begin by briefly recalling the foundational concepts of Hopf
algebras and comodule algebras. Hopf-Galois extensions are introduced, along
with an important correspondence result which relates crossed product algebras
and cleft extensions. Differential calculus over algebras is reviewed, with a
particular focus on the covariant case. The Durdevic theory is presented in a
modern language. We compare this theory to existing literature on the topic. We
extend the Durdevic theory and provide explicit realisations of quantum
principal bundles and complete differential calculi on the noncommutative
algebraic 2-torus, the quantum Hopf fibration and on crossed product algebras,
resulting in an original contribution within this thesis.
