Modalities Through the Looking Glass: A study on coalgebraic modal logic and their applications Raul Andres Leal Abstract: This thesis hovers over the interaction of coalgebras and modal logics. Intuitively, coalgebras are machines from the point of view of the user. They arise from computer science as a promising mathematical foundation for computer systems. Colagebras study different state-based systems, where the set of states can be understood as a black box to which one has limited access. For an intuitive illustration of this, think of a coffee vending machine. Most people do not really know what the inner mechanism of the machine is, or even haven ever seen such mechanism. Nevertheless, they can use the machine efficiently. Here we have an interaction with a system in the black box perspective. The coffee machine is a coalgebra. Modal languages provide an internal, local perspective on relational structures. They origin in philosophy as the informal study of modalities like "it is necessary that...", "it is possible that...", "at some point in the future...". However, thanks to the so-called relational semantics, modal logics have found their way to areas such as linguistics, artificial intelligence, and computer science. This versatility has helped to place modal languages as the appropriate choice of languages to describe coalgebras. Moreover, nowadays, it is also fair to say that modal logics are coalgebraic. This thesis has two parts: Modalities in de Stone age and Coalgebraic modal logics at work. In the first part of this manuscript we investigate coalgebraic modal logics. These logics have become one of the main currents of modal logics for coalgebras. Coalgebraic modal logics bring uniformity to the rising wave of modal logics in computer science and provide generality to the interactions of coalgebras and modal logics. More concretely, the structure of the first part is as follows: we first introduce coalgebraic modal logics as a generalisation of basic modal logic using so-called predicate liftings or concrete modalities. We develop these modalities to introduce the so-called functorial framework for coalgebraic modal logics. This accounts to give an algebraic semantics of modal logics. We then use this perspective to compare different coalgebraic modal logics by means of translations. We devote special attention to the so-called Moss logic. We finish this part with a representation theorem which states that each coalgebraic modal logic within the functorial framework is an axiomatization of a logic of predicate liftings. In the second part of this manuscript we investigate how coalgebraic modal logics can be used to study coalgebras. We work three case studies. In the first case we investigate the use of logics for coalgebras to build coalgebras. More concretely we study the relation between final coalgebras and the Hennessy-Milener property. In the second case we investigate how coalgebraic modal logics align with so-called dynamic logics, dynamic logics are often used to reason about programs. In the third place we study coalgebraic modal logics to describe the ongoing behaviour of a state in a coalgebra. We focus on logics of predicate liftings with fixpoint operators. We give a game semantics and prove a bounded modal property for these logics.