We tend to take turns taking notes. People in our group will often note down particular things of significance (vital NPCs, enemies, locations, etc.), but we also like to make sure everyone is playing with the same deck, as it were.
Myself and another player trade off taking notes on our computers/tablets as the game is progressing, capturing a Cliff Notes version of the game as it happens, then immediately mail it out to the group afterwards. Then later, the DM will take those raw notes and turn it into a more narrative format, which makes for a more exciting read-through for a later date.
But the vital part, the in-game note-taking and immediate post-game e-mails, those have been the most useful.
If your players don't have their electronic devices at the table (I know some DMs ban them), another group I was in had a low-tech solution. The DM had a notebook, and every week someone would volunteer to be the note-taker. There was an XP reward attached to doing this duty, and because of this we were each eager to have our turns so no one person would have a huge amount of extra XP. Part of that duty of note-taking was reading out the notes from last session, to give everyone a recap.