We will focus on a currently intensively developing approach to the logical analysis of questions called Inquisitive Semantics. Although this framework has been explored thoroughly in the last decade, so far not much attention has been paid to its algebraic aspects. However, it turns out that the models of inquisitive semantics generate interesting algebraic structures and it would be desirable to understand better their nature. This paper is intended as a contribution to the algebraic study of inquisitive propositions, i.e. propositions that are expressed not only by declarative sentences but also by questions. We will introduce a class of algebraic structures that are suitable for inquisitive logic and denote them as Inquisitive Algebras. We will explain how questions are represented in these structures (prime elements as declarative propositions, non-prime elements as questions, join as a question-forming operation) and provide several alternative characterizations of inquisitive algebras.