user2266: What are some good resource(s) for learning how bitcoin works?
I can't seem to find any resources that are any more detailed than the basics - how you have wallets, the blockchain, confirmations, mining, etc. The source code for bitcoin is also a little beyond me.
What are some good resources that explain the maths behind bitcoin? i.e. things like:

How to mine bitcoin, step-by-step ('manually' - not how to setup cgminer or GUIMiner)
How to generate a new bitcoin wallet address
How to update the blockchain
etc


user6128: 
first of all nice question.
Secondly, here is Satoshi whitepaper. http://bitcoin.org/bitcoin.pdf I don't know if it's all math you want to know, but might be helpful.
There is also NSA paper, which is probably somehow related. NSA is sometimes said to create bitcoin: http://groups.csail.mit.edu/mac/classes/6.805/articles/money/nsamint/nsamint.htm
I don't know anything more, but you should also consider reading source code after the first link.

user1766: Unfortunately, at this point the best resource is the source code itself. There isn't really good documentation. This is a major issue for the community  and the economy as a whole because it stifles the growth and development of the economy. We all know what happens when only a few people understand and are able to meaningfully manipulate the monetary system . . .

I'd start with the original Satoshi whitepaper: bitcoin.org/bitcoin.pdf 
Then I'd move on to the original Satoshi source code 
And then I'd start swimming through the active development on the reference client (good luck): https://github.com/bitcoin/netspec

The best aids you can find in this quest are people like those here on stackexchange but also in other places such as the Bitcoin IRC channel and the Bitcoin Forums
It would be great for everyone here and the world at large if you documented your learning process on github and the bitcoin wiki. That would be the answer to your question.

user8888: There is a book called Bitcoin Internals that explains how Bitcoin works technically. It covers all the major topics like addresses, transaction processing, the blockchain, mining, pools, etc. 
The Bitcoin wiki (http://bitcoin.it) is a good reference for more specific details. 
Satoshi Nakamoto's whitepaper is also obligatory reading, but it is only a partial explanation of how Bitcoin works.

user6177: Take a look into the technical wiki pages: https://en.bitcoin.it/wiki/Category:Technical
For example, Technical background of version 1 Bitcoin addresses explains how Bitcoin addresses are generated, or the Protocol specification might help you as well.

2266: <p>What are some good resource(s) for learning <strong>how</strong> bitcoin works?</p>

<p>I can't seem to find any resources that are any more detailed than the basics - how you have wallets, the blockchain, confirmations, mining, etc. The source code for bitcoin is also a little beyond me.</p>

<p>What are some good resources that explain the maths behind bitcoin? i.e. things like:</p>

<ul>
<li>How to mine bitcoin, step-by-step ('manually' - not how to setup cgminer or GUIMiner)</li>
<li>How to generate a new bitcoin wallet address</li>
<li>How to update the blockchain</li>
<li>etc</li>
</ul>
 6128: <p><br>
first of all nice question.<br>
Secondly, here is Satoshi whitepaper. <a href="http://bitcoin.org/bitcoin.pdf" rel="nofollow">http://bitcoin.org/bitcoin.pdf</a> I don't know if it's all math you want to know, but might be helpful.<br>
There is also NSA paper, which is probably somehow related. NSA is sometimes said to create bitcoin: <a href="http://groups.csail.mit.edu/mac/classes/6.805/articles/money/nsamint/nsamint.htm" rel="nofollow">http://groups.csail.mit.edu/mac/classes/6.805/articles/money/nsamint/nsamint.htm</a>
<br>I don't know anything more, but you should also consider reading source code after the first link.</p>
 1766: <p>Unfortunately, at this point the best resource is the source code itself. There isn't really good documentation. This is a major issue for the community  and the economy as a whole because it stifles the growth and development of the economy. We all know what happens when only a few people understand and are able to meaningfully manipulate the monetary system . . .</p>

<ul>
<li><p>I'd start with the original Satoshi whitepaper: <a href="http://bitcoin.org/bitcoin.pdf" rel="nofollow">bitcoin.org/bitcoin.pdf</a>  </p></li>
<li><p>Then I'd move on to the original <a href="http://www.bitcointrading.com/forum/bitcoin-clients/original-bitcoin-source-code-archives/" rel="nofollow">Satoshi source code</a>  </p></li>
<li>And then I'd start swimming through the active development on the reference client (good luck): <a href="https://github.com/bitcoin/netspec" rel="nofollow">https://github.com/bitcoin/netspec</a></li>
</ul>

<p>The best aids you can find in this quest are people like those here on stackexchange but also in other places such as the Bitcoin IRC channel and the Bitcoin Forums</p>

<p>It would be great for everyone here and the world at large if you documented your learning process on <a href="http://github.com/bitcoin" rel="nofollow">github</a> and the <a href="http://bitcoin.it" rel="nofollow">bitcoin wiki</a>. That would be the answer to your question.</p>
 8888: <p>There is a book called <a href="http://rads.stackoverflow.com/amzn/click/B00DG8EPT0" rel="nofollow">Bitcoin Internals</a> that explains how Bitcoin works technically. It covers all the major topics like addresses, transaction processing, the blockchain, mining, pools, etc. </p>

<p>The Bitcoin wiki (<a href="http://bitcoin.it" rel="nofollow">http://bitcoin.it</a>) is a good reference for more specific details. </p>

<p>Satoshi Nakamoto's whitepaper is also obligatory reading, but it is only a partial explanation of how Bitcoin works.</p>
 6177: <p>Take a look into the technical wiki pages: <a href="https://en.bitcoin.it/wiki/Category%3aTechnical" rel="nofollow">https://en.bitcoin.it/wiki/Category:Technical</a></p>

<p>For example, <a href="https://en.bitcoin.it/wiki/Technical_background_of_Bitcoin_addresses" rel="nofollow">Technical background of version 1 Bitcoin addresses</a> explains how Bitcoin addresses are generated, or the <a href="https://en.bitcoin.it/wiki/Protocol_specification" rel="nofollow">Protocol specification</a> might help you as well.</p>
