Adventures in Diagonalizable Algebras V. Yu. Shavrukov Abstract: The topic of this dissertation is diagonalizable algebra's, in particular those of formal theories such as Peano arithmetic. Part one discusses algebras that can be embedded in a diagonalizable algebra of a given formal theory. In the case of embeddings with a recursively enumerable range, the set of such algebras is characterized by a small number of simple conditions of an algebraic and recursive nature. In the general case, without restrictions on the range of the embedding, the same holds for theories that are not $\Sigma_1$-correct. In this case, we use Solovay--functions travelling through Kripke models changing with the passage of time. Our constructions are supported by argumentation stemming from modal logic. Amongst others, we prove a uniform variant of the interpolation theorem for the modal logic {\bf L}. Part two is concerned with differences between diagonalizable algebras of Peano arithmetic and Zermelo--Fraenkel set theory. These algebras turn out not to be isomorphic. The proof for this relies on estimates of the length of proofs of specific sentences in these theories. In part three we look at the first-order theories of diagonalizable algebras. Methods from interpretability logic are applied in the construction of an interpretation of the true first-order arithmetic in the first order theory of diagonalizable algebras of $\Sigma_1$-correct formal theories (with the use of results from part one). The interpretation shows that these diagonalizable algebras have an undecidable, even non-arithmetical first-order theory. Furthermore, an alternative approach is developed, that makes use of Post canonical systems, which enables proof of the undecidability of diagonalizable algebras of a broader class of theories.