Band of Brothers is, in my opinion (and my brother's), the greatest mini-series/limited series ever made. Nothing compares. If you enjoy it as much as we do, check out The Pacific. It's a very different vibe, but you might like it as well.

As far as proper TV shows, lotta people have mentioned excellent shows here, but I really want to highlight Deadwood. It's incredible. They built the town as filming progressed, they changed the cursewords of the time to modern ones you could understand how foul it really was, and every single actor knocks it out of the park. Ian McShane, in particular, is a blazing beacon of gravitas.

Honourary mentions:

Oz: It's basically the OG of top-tier television. You'll be surprised how many modern A/B-list actors are in it (same goes for Band of Brothers).

Chernobyl: Probably the closest non-horror horror series I've ever seen. It's a masterpiece.