AbstractLet (𝒳,d, μ) be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that μ(﹛x﹜) = 0 for allx∈ 𝒳. In this paper, we show that the boundedness of a Calderón–Zygmund operatorTonL2(μ) is equivalent to that ofTonLp(μ) for somep∈ (1,∞), and that ofTfromL1(μ) toL1,∞(μ). As an application, we prove that ifTis a Calderón–Zygmund operator bounded onL2(μ), then its maximal operator is bounded onLp(μ) for allp∈ (1,∞) and from the space of all complex-valued Borel measures on 𝒳 toL1,∞(μ). All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.