I can only speak for my job which is leading the video department alongside my close collaborator David, but due to the aspect of very wide number and type of events, we need a lot of different hardware and software tools meet the needs of each type of event.
We have a fleet of video projectors from 3000 to 10 000 lumens and different aspect ratios and full HD screens. We also have plenty of media players like Mac minis, brightsign players, SD Card readers, cameras, streamers etc... And obviously due to the wide type of video signal types and sources with each their pros and cons, we do have a bunch of converters, matrices, extenders and different type of scalers.
With all those type of events and different artistic styles, we can't master every type of technology but we need to understand each project technically speaking and have a wide variety of skills. For instance, we are not motion designers, but we need to know a bit of After Effects to adapt contents to our infrastructures. We are not master coders, but we need to understand code and use it to adapt interactive installations and create tools to help control our devices...
For example, our day to day software tools are VDMX, Touchdesigner, Millumin, Resolume, Modulo Pi media servers Max/MSP, openFrameworks, Processing, Arduino... and protocols such as OSC, MIDI or NDI.