I use GNU Cash and it uses Open Financial Exchange (ofx) http://www.ofx.net/ to download complete transactions and balances from each account of each bank.
Let me emphasize that again, you get a huge list of transactions with OFX into the GNU Cash. Depending on the account type these transactions can be very detailed description of your transactions (purchases+paycheques), investments, interests, etc.
In my case, even though I have Chase debit card I had to choose Chase Credit to make it work. But Chase wants you to enable this OFX feature by logging into your online banking and enable Quicken/MS Money/etc. somewhere in your profile or preferences. Don't call Chase customer support because they know nothing about it.
This service for OFX and GNU Cash is free. I have heard that they charge $10 a month for other platforms.
OFX can download transactions from 348 banks so far. http://www.ofxhome.com/index.php/home/directory
Actualy, OFX also supports making bill payments, stop a check, intrabank and interbank transfers etc. It is quite extensive. See it here: http://ofx.net/AboutOFX/ServicesSupported.aspx