The paper discusses a class of $f(R)$ gravity models where the function $f(R)$ is a function of the Ricci scalar $R$. These models produce viable cosmology different from the Lambda-CDM model at recent times and satisfy cosmological, solar system, and laboratory tests. The models have both flat and de Sitter space-times as particular solutions in the absence of matter. In flat space-time, the cosmological constant is zero, but it appears effectively in a curved space-time for sufficiently large $R$. A key feature is that the cosmological constant "disappears" in flat space-time, which could be detected through small discrepancies in the slope of the primordial perturbation power spectrum from galaxy surveys and CMB fluctuations. However, a new problem arises in these models: the possible overproduction of new massive scalar particles (scalarons) in the early universe. The paper also explores the stability conditions and the behavior of the model's FRW solutions, laboratory and solar system tests, and the dynamics of small perturbations. The conclusion highlights the potential of these models to explain dark energy but also points out the need to address the issue of scalaron overproduction.The paper discusses a class of $f(R)$ gravity models where the function $f(R)$ is a function of the Ricci scalar $R$. These models produce viable cosmology different from the Lambda-CDM model at recent times and satisfy cosmological, solar system, and laboratory tests. The models have both flat and de Sitter space-times as particular solutions in the absence of matter. In flat space-time, the cosmological constant is zero, but it appears effectively in a curved space-time for sufficiently large $R$. A key feature is that the cosmological constant "disappears" in flat space-time, which could be detected through small discrepancies in the slope of the primordial perturbation power spectrum from galaxy surveys and CMB fluctuations. However, a new problem arises in these models: the possible overproduction of new massive scalar particles (scalarons) in the early universe. The paper also explores the stability conditions and the behavior of the model's FRW solutions, laboratory and solar system tests, and the dynamics of small perturbations. The conclusion highlights the potential of these models to explain dark energy but also points out the need to address the issue of scalaron overproduction.