As previously stated, arguably the most mature and widely used set of tools is currently a combination of Pymatgen, FireWorks, Custodian, and Atomate (which is built upon the prior three Python packages). These tools were constructed as part of the Materials Project but have seen uses in other high-throughput DFT studies.
Another general workflow package for automating high-throughput DFT calculations is AFLOW, which has been used in constructing the AFLOWlib repository.
A similar package is qmpy, which has been used in constructing the Open Quantum Materials Database. As you can see, with each new database of DFT-computed properties, there are often specific workflow packages associated with them (typically because everyone's preferences and use-cases are different).
One of the benefits of the AiiDA package mentioned in the prior answer here is that it retains the calculation history of the entire workflow. Seeing as most robust workflows are somewhat dynamic in their settings (e.g. if an error appears, settings are changed), this can be useful for ensuring full transparency and reproducibility. AiiDA is what powers the data on the Materials Cloud.
There are also many field-specific packages that attempt to automate workflows specific to that field, oftentimes using one or more of the aforementioned packages. For instance, the Generalized Adsorption Simulation for Python (GASpy) code by the Ulissi group is well-suited for automating DFT calculations of inorganic surfaces, as outlined here. Rosen et al. have also developed a workflow for automating DFT calculations of metal–organic frameworks, as described here. Yet another is the MAterials Simulation Toolkit (MAST), which allows users to build up "recipes" for automated workflows in a similar manner as Atomate and was originally developed with a focus on simulating defects and diffusion in solids, as described here.
Edit #1: A new package, named [AMP$^2$], was just published for automating DFT calculations of crystals. It looks like it has several ease-of-use features, such as robust default settings for commonly computed properties, automatically testing if a hybrid functional should be used, and an algorithm to identify complex magnetic orderings in an automated fashion. The code is available to download here.
Edit #2: I somehow forgot to mention that the Atomic Simulation Environment (ASE), which some of the prior toolkits use, is an extremely useful and flexible resource that can be used to construct your own workflows with some Python scripting. It's not specifically meant for high-throughput but can be used that way.