I\'m sure that the answer to this question is No, but I can\'t seem to find a way that simply transforming < and > to < and <
No, it's not sufficient. Remember that XSS isn't just about untrusted data in HTML, you'll also find it in JavaScript and CSS. Think about a situation such as "var myVar = [input];" There are all sorts of malicious things you can do with that [input] value without going anywhere near angle brackets. There's many more examples over in the XSS cheat sheet: http://ha.ckers.org/xss.html
You've mentioned ASP.NET in the tag; what you want to be looking at is the [AntiXSS library][1]. Grab this and use the appropriate output encoding:
Encoder.CssEncode()
Encoder.HtmlEncode()
Encoder.HtmlAttributeEncode()
Encoder.JavaScriptEncode()
etc. etc. There's absolutely no reason to try and do your own character substitution in .NET.
Not all XSS attacks include < or > at all, depending on where the data is being inserted.
https://www.owasp.org/index.php/XSS_(Cross_Site_Scripting)_Prevention_Cheat_Sheet#Why_Can.27t_I_Just_HTML_Entity_Encode_Untrusted_Data.3F
When using an untrusted string in an attribute (quoted with ") you need to escape " as ".
Otherwise you could easily inject javascript. For example, <a href="{{str}}"> with str being, for example, " onmouseover='something-evil'".
No. Here are a couple of examples where escaping <, >, ', " and & is not enough:
Example 1:
<a href="{{myUrl}}">
XSS Attack:
myUrl = "javascript:alert(1)"
Example 2:
<script>var page = {{myVar}};</script>
XSS Attack:
myVar = "1;alert(1)"
See https://www.owasp.org/index.php/XSS_(Cross_Site_Scripting)_Prevention_Cheat_Sheet for ways of preventing these attacks.