  We present a device for specifying and reasoning about syntax for datatypes,
programming languages, and logic calculi. More precisely, we study a notion of
"signature" for specifying syntactic constructions.
  In the spirit of Initial Semantics, we define the "syntax generated by a
signature" to be the initial object -- if it exists -- in a suitable category
of models. In our framework, the existence of an associated syntax to a
signature is not automatically guaranteed. We identify, via the notion of
presentation of a signature, a large class of signatures that do generate a
syntax.
  Our (presentable) signatures subsume classical algebraic signatures (i.e.,
signatures for languages with variable binding, such as the pure lambda
calculus) and extend them to include several other significant examples of
syntactic constructions.
  One key feature of our notions of signature, syntax, and presentation is that
they are highly compositional, in the sense that complex examples can be
obtained by gluing simpler ones. Moreover, through the Initial Semantics
approach, our framework provides, beyond the desired algebra of terms, a
well-behaved substitution and the induction and recursion principles associated
to the syntax.
  This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi,
which, in turn, was directly inspired by some earlier work of
Ghani-Uustalu-Hamana and Matthes-Uustalu.
  The main results presented in the paper are computer-checked within the
UniMath system.
