  In this article we introduce the variety of monadic BL-algebras as
BL-algebras endowed with two monadic operators $\forall$ and $\exists$. After a
study of the basic properties of this variety we show that this class is the
equivalent algebraic semantics of the monadic fragment of H\'ajek's basic
predicate logic. In addition, we start a systematic study of the main
subvarieties of monadic BL-algebras, some of which constitute the algebraic
semantics of well-known monadic logics: monadic G\"odel logic and monadic
{\L}ukasiewicz logic. In the last section we give a complete characterization
of totally ordered monadic BL-algebras.
