We study Banach \(C(K)\) -spaces which are Grothendieck. Recall that a Banach space E is a Grothendieck space (or has the Grothendieck property) if every weak* convergent sequence in the dual space \(E'\) is weakly convergent. Two characterizations of Grothendieck spaces \(C(K)\) will be provided: in terms of operators onto the Banach space \(c_0\) and in terms of sequences of Radon measures related to the classical Josefson–Nissenzweig theorem from Banach space theory. We also present in detail a proof of Grothendieck’s theorem asserting that for every extremely disconnected compact space K the space \(C(K)\) is Grothendieck—for this purpose we present proofs of Rosenthal’s “disjointification” lemma for sequences of measures and Dieudonné–Grothendieck’s characterization of weakly compact subsets of dual spaces \(C(K)^{\prime }\) .