In this thesis, we present some important examples of reconstruction theorems, i.e. theorems that give correspondence between two objects in the sense that one object can be completely reconstructed from the other. An important class of reconstruction theorems is duality between geometric spaces and function algebras. Included here are Stone duality, Gel’fand-Naimark theorem and reconstruction of algebraic sets, which are covered in chapters 2, 3 and 4. Apart from spaces, we can also reconstruct some other geometric objects, such as sheaves and vector bundles. Affine Serre theorem from chapter 5 is an example of such reconstruction. All of these theorems are stated in the language of category theory, which is why the first chapter is dedicated to basic definitions and results of category theory. Also, category theory provides us with a new formulation of algebraic geometry in which spaces are described as representable functors. This so-called functorial geometry enables us to transfer to the noncommutative case easily. This way we are able to define a quantum space and a quantum group, which is an automorphism group of a quantum space. Another important class of reconstruction theorems refers to reconstructing symmetry objects such as automorphism groups. We cover them in chapter 7. These theorems are known by their common name Tannaka duality, and they describe reconstructions of groups, groupoids, Hopf algebras and similar objects from their categories of representations. The last chapter gives an overview of some common ideas from the theorems of the preceding chapters. We present the development of the notion of spectrum and explain the importance of these dualities for noncommutative geometry.