We study the notion of frequent hypercyclicity that was recently introduced by Bayart and Grivaux [1], [2]. We show that frequently hypercyclic operators satisfy the Hypercyclicity Criterion, answering a question of Bayart and Grivaux [2]. We also disprove a conjecture in [2] concerning frequently hypercyclic weighted shifts, and we prove that vectors which have a somewhere frequently dense orbit are frequently hypercyclic.