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The Multi-Kink Quantile Regression (MKQR) model is an important tool for analyzing data with heterogeneous conditional distributions, especially when quantiles of response variable are of interest, due to its robustness to outliers and heavy-tailed errors in the response. It assumes different linear quantile regression forms in different regions of the domain of the threshold covariate but are still continuous at kink points. In this paper, we investigate parameter estimation, kink point detection and statistical inference in MKQR models. We propose an iterative segmented quantile regression algorithm for estimating both the regression coefficients and the locations of kink points. The proposed algorithm is much more computationally efficient than the grid search algorithm and not sensitive to the selection of initial values. Asymptotic properties, such as selection consistency of the number of kink points, asymptotic normality of the estimators of both regression coefficients and kink effects, are established to justify the proposed method theoretically. A score test, based on partial subgradients, is developed to verify whether the kink effects exist or not. Test-inversion confidence intervals for kink location parameters are also constructed. Intensive simulation studies conducted show the proposed methods work very well when sample size is finite. Finally, we apply the MKQR models together with the proposed methods to the dataset about secondary industrial structure of China and the dataset about triceps skinfold thickness of Gambian females, which leads to some very interesting findings. A new R package MultiKink is developed to implement the proposed methods.