  We introduce the notion of differential largeness for fields equipped with
several commuting derivations (as an analogue to largeness of fields). We lay
out the foundations of this new class of "tame" differential fields. We state
several characterizations and exhibit plenty of examples and applications. Our
results strongly indicate that differentially large fields will play a key role
in differential field arithmetic. For instance, we characterise differential
largeness in terms of being existentially closed in their power series field
(furnished with natural derivations), we give explicit constructions of
differentially large fields in terms of iterated powers series, we prove that
the class of differentially large fields is elementary, and we show that
differential largeness is preserved under algebraic extensions, therefore
showing that their algebraic closure is differentially closed.
