Overview of the different types of mixing sequences

Question

I have come across the terms strong mixing, $\alpha$-mixing, $\beta$-mixing, $\phi$-mixing, $\rho$-mixing. Could somebody please compile an answer that would summarize their definition in some clear and unified way, the relationships between these different types of mixing, some intuition about them, their application domain, or why they are natural/interesting to consider. This question aims at becoming a starting point for anyone interested in this topic.

Answer 1

Bradley (2005) "Basic Properties of Strong Mixing Conditions: A Survey and Some Open Questions" and his subsequent entry "Strong mixing conditions" in the Springer Encyclopedia of Mathematics are very readable introductions to the subject. The core idea is that there are processes where as time elapses, future events can be regarded as "increasingly independent" from the past events. The different definitions of mixing refer to different formalisations of "increasingly independent".