First check that all your settings are correct, that reminders syncing (calendars on a computer) is checked on all devices (system preferences > iCloud on a mac and settings > iCloud on a iPhone, iPad or iPod).
Make sure the reminders you are adding are added to your 'iCloud' account and not an 'On My Mac', 'On My Phone' or other non iCloud account (you can do this by checking in accounts on an iOS device by tapping the icon with three lines in the top left corner of the screen, or the right side panel by clicking on the small triangle to the right of the word 'Reminders' in iCal on a computer), non iCloud reminders will not sync.
If you are sure that everything is set up correctly and your reminders are in the iCloud group, you might try unchecking reminders syncing (calendars syncing on a computer) in the iCloud settings, restarting your device and then re-enabling notes syncing.
Thank you Winston Churchill for your advices : I absolutelly followed ALL your tips, but nothing change :-(
The data displayed on icloud.com for my account are obsolte about Reminders and now I have the same problem with my Calendars ! (but not with Notes wich functions).
I can add some comments about the behavior with Reminders :
1) After login on icloud.com : Firstly I see all the icons for the different applications (Mail, Calendar, Reminders, Notes, Find my iPhone, etc...) without badges.
2) Then just few seconds later, a badge containing the number '1' appear in front of the Reminder icon. This value seems correct because I have the same value in the badge on Mountain Lion's Reminder icon.
3) When I click on the Reminder icon on icloud.com, immediately appear a loading spinner and the value inside the badge change to '3'. The '3' value seems incorrect.
4) Then appear the Reminder page with obsolete data including 3 late reminders and the reminder lists are incomplete, I have created one more shared list a few weeks ago.
To put it in a nutshell, I regret that iCloud.com isn't reliable for me. Too much old datas via the website.
I hope Apple will quickly fix theses bugs about icloud.com sync for some people like me.