The paper investigates systematically the Hardy spaces \(HA_{p, q}\) (\(H\dot A_{p,q}\)) associated to Herz spaces \(A_{p,q}\) \((\dot A_{p,q})\). The main result is Theorem 2.14. Let \({n- 1\over n}< q\leq 1< p< \infty\), \(f\in {\mathcal S}'(R)\). Then the following properties concerning the Hardy-Herz spaces are equivalent: 1. There is \(F(x, t)= (u_0(x),\dots, u_n(x))\), a Stein-Weiss system of harmonic functions such that \(|||F(\cdot, t)|||_{\dot A_{p, q}}< \infty\), \(|F(x, t)|= \left(\sum^n_{j= 0} |u_j(x, t)|^2\right)^{1/2}\), and \(u_0(x)\to f(x)\), \(t\to 0\), in the distribution sense; 2. The vertical maximal function \(m^+_u(x)\in \dot A_{p,q}\), where \(u(x)\) is the harmonic extension of \(f\), and \(u(x,t)\to f(x)\), \(t\to 0\); 3. The non-tangential maximal function \(m_u(x)\in \dot A_{p,q}\) (with the amplitude 1); 4. \(m^N_u(x)\in \dot A_{p,q}\), with the amplitude \(N\); 5. The grand maximal function \(G(f)\in \dot A_{p, q}\)' 6. \(f(x)= \sum \lambda_k a_k(x)\), \(a_k\)'s central \((p, q, s)\) atoms, \(N\geq \left[{1\over n}\left({1\over q}- 1\right)\right]\) and \(\sum|\lambda_k|^q< \infty\). In addition, the paper gives the Littlewood-Paley function characterization of \(HA_{p,q}\) \((H\dot A_{p,q})\) for \(0< q\leq 1< p\leq 2\), and the dual spaces of \(HA_{p, q}\) \((H\dot A_{p,q})\), and the complex interpolation of \(HA_{p,q}\) \((H\dot A_{p, q})\).