BACKGROUND: In meta-analysis, the normal distribution assumption has been adopted in most systematic reviews of random-effects distribution models due to its computational and conceptual simplicity. However, this restrictive model assumption is possibly unsuitable and might have serious influences in practices. METHODS: We provide two examples of real-world evidence that clearly show that the normal distribution assumption is explicitly unsuitable. We propose new random-effects meta-analysis methods using five flexible random-effects distribution models that can flexibly regulate skewness, kurtosis and tailweight: skew normal distribution, skew t-distribution, asymmetric Subbotin distribution, Jones-Faddy distribution, and sinh-arcsinh distribution. We also developed a statistical package, flexmeta, that can easily perform these methods. RESULTS: Using the flexible random-effects distribution models, the results of the two meta-analyses were markedly altered, potentially influencing the overall conclusions of these systematic reviews. CONCLUSIONS: The restrictive normal distribution assumption in the random-effects model can yield misleading conclusions. The proposed flexible methods can provide more precise conclusions in systematic reviews.