  We establish a complete theory of the flag Hardy space on the Heisenberg
group $\mathbb H^{n}$ with characterisations via atomic decompositions, area
functions, square functions, maximal functions and singular integrals. We
introduce several new techniques to overcome the difficulties caused by the
noncommutative Heisenberg group multiplication, and the lack of a suitable
Fourier transformation and Cauchy--Riemann type equations. Applications include
the boundedness from the flag Hardy space to $L^1(\mathbb H^n)$ of various
singular integral operators that arise in complex analysis, a sharp boundedness
result on the flag Hardy space of the Marcinkiewicz-type multipliers introduced
by M\"uller, Ricci and Stein, and the decomposition of flag BMO space via
singular integrals.
