Almost exactly a year ago, I wrote a blog post titled Solving the Emoji/character encoding problem in Drupal 7.
Since writing that post, Drupal 7 bugfixes and improvements have started to pick up steam as (a) many members of the community who were focused on launching Drupal 8 had time to take a breather and fix up some long-standing Drupal 7 bugs and improvements that hadn't yet been backported, and (b) there are two new D7 core maintainers. One of the patches I've been applying to many sites and hoping would get pulled into core for a long time was adding support for full UTF-8, which allows the entry of emojis, Asian symbols, and mathematical symbols that would break Drupal 7 sites running on MySQL previously.
My old blog post had a few steps that you could follow to make your Drupal 7 site 'mostly' support UTF-8, but there were some rough edges. Now that support is in core, the process for converting your existing site's database is more straightforward: