Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)In many studies, limited or censored data are collected. This occurs, in several practical situations, for reasons such as limitations of measuring equipment or from experimental design. Hence, the exact true value is recorded only if it falls within an interval range, so, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data. Most of these models are based on the normality assumption for the error terms. However, such analyses might not provide robust inference when the normality assumption (or symmetry) is questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumption for the random errors with the asymmetric class of scale mixtures of skew-normal (SMSN) distributions. The SMSN is an attractive class of asymmetrical heavy-tailed densities that includes the skew-normal, skew-t, skew-slash, the skew-contaminated normal and the entire family of scale mixtures of normal distributions as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo (MCMC) algorithm is introduced to carry out posterior inference. The likelihood function is utilized to compute not only some Bayesian model selection measures, but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measures. The proposed Bayesian methods are implemented in the R package BayesCR, proposed by us. The newly developed procedures are illustrated with applications using real and simulated data.103425439Sao Paulo State Research Foundation (FAPESP) [2012/18702-9, 2013/21468-0, 2014/02938-9]CNPq [308243/2012-9, 447964/2014-3]FAPEAM [0390/2013]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)