How do I count occurrences by day in SQL?

Newbie SQL question here -->

I've got an Occurrences table that contains a row for each time a user did something. A user can do the thing multiple times per day. It looks like this:

Date      Username
------    --------
1/1/9     User1
1/1/9     User1
1/1/9     User2
1/2/9     User1
1/2/9     User3
1/3/9     User1
1/3/9     User1
1/3/9     User1
1/3/9     User2
1/3/9     User3

I want a query to simply return the Username and the quantity (count) of unique days they appear. For the above data set, the result I'm looking for would be:

Username    UniqueDaysAppeared
--------    ------------------
User1       3
User2       2
User3       2

I keep getting screwed up because my query is returning not the count of unique days per user but rather the number of occurrences of the user overall.