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Two fundamental research tasks in science and engineering are forward predictions and data inversion. This article introduces a recent R package RobustCalibration for Bayesian data inversion and model calibration by experiments and field observations. Mathematical models for forward predictions are often written in computer code, and they can be computationally expensive slow to run. To overcome the computational bottleneck from the simulator, we implemented a statistical emulator from the RobustGaSP package for emulating both scalar-valued or vector-valued computer model outputs. Both posterior sampling and maximum likelihood approach are implemented in the RobustCalibration package for parameter estimation. For imperfect computer models, we implement Gaussian stochastic process and the scaled Gaussian stochastic process for modeling the discrepancy function between the reality and mathematical model. This package is applicable to various types of field observations, such as repeated experiments and multiple sources of measurements. We discuss numerical examples of calibrating mathematical models that have closed-form expressions, and differential equations solved by numerical methods.