A rather flexible approach to multi-task learning consists in solving a regularization problem where a suitable kernel is used to model joint relationships between both inputs and tasks. Since specifying an appropriate multi-task kernel in advance is not always possible, estimating one from the data is often desirable. Herein, we overview a class of techniques for learning a multi-task kernel that can be decomposed as the product of a kernel on the inputs and one on the task indices. The kernel on the task indices (output kernel) is optimized simultaneously with the predictive function by solving a joint two-level regularization problem.