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Real data are unlikely to be exactly normally distributed. For better estimation and inference with non-normal data, I work on multiple projects in recent years. In terms of estimation, my collaborators and I propose a distributionally weighted least squares (DLS) estimator for parameter estimation (Du & Bentler, 2022; Du et al., 2022). We find that DLS works well with both normal and non-normal data in both factor analysis and growth curve models. We currently are exploring different ways to estimate DLS including bootstrapping and asymptotic methods. In terms of model fit inference, we propose to use an unbiased distribution free weight matrix estimator in robust test statistics and extend it to models with mean structures (Du & Bentler, 2022; Du, 2022). We find that the Satorra–Bentler statistic and Haya- kawa statistic coupled with unbiased distribution free weight matrix could control Type I error rates better than other statistics.