Let \(\left( {\mathcal{Y},d,d\lambda } \right)\) be (ℝn, |·|, µ), where |·| is the Euclidean distance, µ is a nonnegative Radon measure on ℝn satisfying the polynomial growth condition, or the Gauss measure metric space (ℝn, |·|, dλ), or the space (S, d, ρ), where S ≡ ℝn ⋉ ℝ+ is the (ax + b)-group, d is the left-invariant Riemannian metric and ρ is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces \(\left\{ {X_s \left( \mathcal{Y} \right)} \right\}_{0 < s \leqslant \infty }\) and the BMO-type spaces \(\left\{ {BMO\left( {\mathcal{Y}, s} \right)} \right\}_{0 < s \leqslant \infty }\). Let H1\(\left( \mathcal{Y} \right)\) be the known atomic Hardy space and L01\(\left( \mathcal{Y} \right)\) the subspace of f ∈ L1\(\left( \mathcal{Y} \right)\) with integral 0. The authors prove that the dual space of Xs\(\left( \mathcal{Y} \right)\) is \(BMO\left( {\mathcal{Y},s} \right)\) when s ∈ (0,∞), Xs\(\left( \mathcal{Y} \right)\) = H1\(\left( \mathcal{Y} \right)\) when s ∈ (0, 1], and X∞\(\left( \mathcal{Y} \right)\) = L01\(\left( \mathcal{Y} \right)\) (or L1\(\left( \mathcal{Y} \right)\)). As applications, the authors show that if T is a linear operator bounded from H1\(\left( \mathcal{Y} \right)\) to L1\(\left( \mathcal{Y} \right)\) and from L1\(\left( \mathcal{Y} \right)\) to L1,∞\(\left( \mathcal{Y} \right)\), then for all r ∈ (1,∞) and s ∈ (r,∞], T is bounded from Xr\(\left( \mathcal{Y} \right)\) to the Lorentz space L1,s\(\left( \mathcal{Y} \right)\), which applies to the Calderon-Zygmund operator on (ℝn, |·|, µ), the imaginary powers of the Ornstein-Uhlenbeck operator on (ℝn, |·|, dγ) and the spectral operator associated with the spectral multiplier on (S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.