Unicode support is reasonably universal by now in browsers. However, this does not mean that all characters are supported. The Unicode Standard does not require that any specific collection of characters be supported. No font covers all of Unicode, and the odds of finding an implementation where even the union of installed fonts covers all of Unicode are infinitesimally small. Well, very small at least.
If you just need to have characters rendered, then the main problem is with fonts, though some browsers are less capable of using all the installed fonts than others. If you need other support, such as correct right-to-left rendering when needed, or Unicode-style line breaking rules, then it’s more an issue with browsers.
To summarize, you probably need to refine your question considerably in order to get practically useful answers.