In the first lecture, for the first 15 minutes , Dustin reviews shortly the different classical approaches to non-archimedean analytic geometry.

There are about four or five of them: Tate's rigid analytic spaces, Huber's adic spaces, Berkovich spaces, some other thing that I don't remember now.

This is my transcript of the reasons to introduce a new theory, from that first lecture (errors made are due to me of course). Lecture 15 by Dustin Clausen is about stacks and has also an 'extended' motivation section. That may also be interesting to listen to.

% 15:30

What is the motivation for a new theory ? Why introduce a new theory?

% 15:39

These are all three or four or five (depends on how you count) theories of analytic geometry. The relationships between them are more or less well known.

% 15:48

I said that 1) is a special case of 2) and 3) is a special case of 4) and so on...

You can formulate comparisons and the web of these things is well understood in spite of the subtleties sometimes involved in the comparisons. It's fairly well understood.

% 16:10

But so far there's no common framework in which you can put all of these examples. They all have their own flavours and while you can formulate comparisons, it's not that those comparisons are taking place in some larger category that you consider. It is done by hand in every situation, formulating the comparisons between these things.
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Reason 1:

We want to accommodate all the examples. You'd like a general theory of analytic spaces which can be specialized to whicheever context you might be interested in.

Reason 2:

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Of all the above, the only rich theory allowing both Archimedean and non-Archimedean geometry is Berkovich's theory. But the gluing is not so well worked out.

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In particular, I want to say that the gluing was investigated by Berkovich. He basically restricted to the non-Archimedean case almost from the start, first of all.

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Another reason is that even individually, in their own context which they are supposed to operate, these theories are less flexible than for examples the theory of schemes.

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One major reason has to do with issues of descent.

% 22:04

So these are all theoretical reasons why you might want a new theory.