Least squares support vector regression (LS-SVR) is a robust machine learning algorithm for small sample data. Its solution is derived from solving a set of linear equations, making the calculation process straightforward. In order to overcome the difficulties of the regression estimations when the responses are subject to interval censoring or left truncation and right censoring, two LS-SVR methods are proposed. For interval-censored data, one can easily estimate the regression functions by combining the imputation techniques and LS-SVR for right-censored data. For left-truncated and right-censored data, a weight is used to reduce the effects of truncation and censoring on the LS-SVR procedure. Simulation results show that the proposed methods can reduce regression error and yield high accuracy and stability.