Given the large number of papers written over the last ten years on error controls in high dimensional multiple testing, it would be worthwhile to consider a single comprehensive technique that allows user flexibility in error control when dealing with big-data. We describe a new and comprehensive family of error rates that contains and generalizes most existing proposals. It offers the scientist a broad choice on how to properly control for discovering false findings. We also propose a corresponding family of control procedures that guarantees the control of the new error rates under different assumptions on the p-values. We show the interest of introducing this comprehensive error rate to obtain new interesting theoretical results on assumption weakening, relation between different error rates and on asymptotic control. We also discuss some particular choices of error rates that bridge the gap between two well-known control error metrics: FWER and FDR. The comprehensive family and the corresponding control theorems open new perspectives in the field of multiple testing.