There is a quest journal! That was the big discovery today, and it took some digging around to find it   Quests seem like personal things to me, so I looked for quest log of some sort in the unit info window. It seems quests are global though, any unit can finish a quest another champion started? The placement of the quest journal in the kingdom info window suggests this is also how it's meant to be later on too. If that's so then it's logical but it would help with something that leads my attention to the quest view from the champions or from the quest targets on the map. Perhaps clicking the target in the world shows a button that opens the relevant quest in the journal view?

Anywho, here are my quick first impressions on the current quest log:

• Navigation is way too small and discrete. It took me a long while of pixel hunting before I saw the tiny arrow in the bottom right corner. That arrow is also very flat and doesn't look clickable. Make it larger and give it a slightly more button-like look, just some discreet shadowing to make it look more like a pushable button.  
• There's no UI control to separate active from finished quests. It says in the description if they are active or not and it seems the active ones always come first (not sure though), but it's a discreet description in italics that would be better built into the interface itself. Perhaps as tabs, something that sorts quests into the two categories. If I want to check what happened in the last quest completed I don't want to have to click through five active ones first. 
• There's a lot of unused space in the view. Not a big problem in itself but lends itself to optimization of use of space. Given the current short descriptions of the quests and the size of the illustration, perhaps it's possible to fit several quests on top of each other and just give them discrete frames to separate them? The selected quest could then be the one with info and illustration shown. This could be an alternative to both issues mentioned above, you could instead have a long list of quests with categories Active and Complete where the selected one is the only with info shown, the others only show title. Click one and you see it's description and illustration instead.