Having tried multiple campaigns over the years, they can be a LOT of fun. They can also be a logistical nightmare in a group like ours.

Back when I tried running a big campaign for WFB, half of the players had officially dropped out, failed to respond or avoided joining any battles by the middle of the campaign, and that was with a two month window per turn. I'd like to think that those players who stuck it out to the end had fun though.
There are obviously lots of options to run a campaign, so a lot depends on what you're looking for. The campaigns I have seen/played have included:
Simply abstracting the armies on the maps and concentrating on the strategic action, effectively making the game a board game, somewhere between Risk and just clicking auto-resolve in Warhammer Total War.
You can go for the full on army management and tactical deployments where each combat on the map is played out as it occurs. Characters develop skills, units develop characteristics, warriors rise through the ranks to become legends etc. This allows for the narrative style of events M_i_J mentions above.
Or you can use something like the current 40K rules where you count overall victories for a particular faction and while units do get upgrades over time, you never really get the asymmetrical battles of the first two styles.
Finally you can go for the full Kriegspiel campaign where you have two teams of generals issuing orders to their units via a GM, who tallies up results and reports back to the generals. At no point does either team have complete awareness of where enemy forces are, or how strong they are. The GM can give teams intelligence reports based on their actions, but you are commanding with a heavy Fog of War effect.
Of course, there are combinations of all of the above and variants of each.
And even once you know what style you want to use, getting everyone involved and keeping them focused on it is a challenge in and of itself.