user12039: I know you can use various APIs to get the balance for a given bitcoin address, but is there a way to total all the bitcoin that were ever received at a given address?
A so-called "transaction history" for a given address?

user2306: Bitcoin-Abe can do this, and I wrote a guide on installing it here.
Once you have it installed, issue a getrecievedbyaddress request.

user21394: BlockTrail's address API will do what you're looking for.
Calling URL:
https://api.blocktrail.com/v1/btc/address/1NcXPMRaanz43b1kokpPuYDdk6GGDvxT2T?api_key=MY_APIKEY

Result:
{
    "address": "1NcXPMRaanz43b1kokpPuYDdk6GGDvxT2T",
    "hash160": "ED12908714FFD43142BF9832692017E8AD54E9A8",
    "balance": 49497392,
    "received": 115015000,
    "sent": 65517608,
    "unconfirmed_received": 0,
    "unconfirmed_sent": 0,
    "unconfirmed_transactions": 0,
    "total_transactions_in": 4,
    "total_transactions_out": 10,
    "category": "donations",
    "tag": "blocktrail"
}


12039: <p>I know you can use various APIs to get the balance for a given bitcoin address, but is there a way to total all the bitcoin that were ever received at a given address?</p>

<p>A so-called "transaction history" for a given address?</p>
 2306: <p>Bitcoin-Abe can do this, and I wrote a guide on installing it <a href="https://bitcoin.stackexchange.com/questions/5518/how-do-i-access-information-on-blockexplorer-com-or-blockchain-info-without-issu/5519#5519">here</a>.</p>

<p>Once you have it installed, issue a <code>getrecievedbyaddress</code> request.</p>
 21394: <p><a href="https://www.blocktrail.com/api/docs#api_address" rel="nofollow">BlockTrail's address API</a> will do what you're looking for.</p>

<p>Calling URL:</p>

<pre><code>https://api.blocktrail.com/v1/btc/address/1NcXPMRaanz43b1kokpPuYDdk6GGDvxT2T?api_key=MY_APIKEY
</code></pre>

<p>Result:</p>

<pre><code>{
    "address": "1NcXPMRaanz43b1kokpPuYDdk6GGDvxT2T",
    "hash160": "ED12908714FFD43142BF9832692017E8AD54E9A8",
    "balance": 49497392,
    "received": 115015000,
    "sent": 65517608,
    "unconfirmed_received": 0,
    "unconfirmed_sent": 0,
    "unconfirmed_transactions": 0,
    "total_transactions_in": 4,
    "total_transactions_out": 10,
    "category": "donations",
    "tag": "blocktrail"
}
</code></pre>
