iOS application: how to clear notifications?

It might also make sense to add a call to clearNotifications in applicationDidBecomeActive so that in case the application is in the background and comes back it will also clear the notifications.

- (void)applicationDidBecomeActive:(UIApplication *)application
{
    [self clearNotifications];
}

Just to expand on pcperini's answer. As he mentions you will need to add the following code to your application:didFinishLaunchingWithOptions: method;

[[UIApplication sharedApplication] setApplicationIconBadgeNumber: 0];
[[UIApplication sharedApplication] cancelAllLocalNotifications];

You Also need to increment then decrement the badge in your application:didReceiveRemoteNotification: method if you are trying to clear the message from the message centre so that when a user enters you app from pressing a notification the message centre will also clear, ie;

[[UIApplication sharedApplication] setApplicationIconBadgeNumber: 1];
[[UIApplication sharedApplication] setApplicationIconBadgeNumber: 0];
[[UIApplication sharedApplication] cancelAllLocalNotifications];

Most likely because Notification Center is a relatively new feature, Apple didn't necessarily want to push a whole new paradigm for clearing notifications. So instead, they multi-purposed [[UIApplication sharedApplication] setApplicationIconBadgeNumber: 0]; to clear said notifications. It might seem a bit weird, and Apple might provide a more intuitive way to do this in the future, but for the time being it's the official way.

Myself, I use this snippet:

[[UIApplication sharedApplication] setApplicationIconBadgeNumber: 0];
[[UIApplication sharedApplication] cancelAllLocalNotifications];

which never fails to clear all of the app's notifications from Notification Center.