I love visiting sites that are innovative and magical. Most of these sites use Flash in most, if not all, their magic.
Flash, if not done right, can create many problems for your visitors. Flash programs can be purchased separately for your website from vendors that specialize in Flash. The programming for your specific website can become fairly complicated, adding additional expense to your site.
All visitors may not be able to view your site as designed so back up code for visitors with older browsers or current browsers that don't read the code needs to be included, making this a job not to be recommended for a DIY job.
Another issue to keep in mind when adding Flash to your website is that Flash is not searchable by search engines. Special care should be taken with Search Engine Optimization so that your site gets the recognition it deserves.