In this study, we introduce a new model called the Odd Burr Lindley distribution which extends the Lindley distribution and has increasing, bathtub and upside down shapes for the hazard rate function. It includes the odd Lindley distribution as a special case. Several statistical properties of the distribution are explored, such as the density, hazard rate, survival, quantile functions, and moments. Estimation using the maximum likelihood and inference of a random sample from the distribution are investigated. A simulation study is performed to compare the performance of the different parameter estimates in terms of bias and mean square error. Two real data applications are modelled with the proposed distribution to illustrate the performance of the new distribution. Based on goodness-of-fit statistics, the new distribution outperforms the generalized gamma, gamma Weibull, gamma exponentiated exponential, generalized Lindley, Kumaraswamy Lindley, and odd log-logistic Lindley distributions.