Android - Access Android device's file system over Bluetooth or Wi-Fi

Over Wi-Fi
Install a small app SwiFTP FTP Server. (original link 'dead' - 2012-09-22, see 'http://ppareit.github.com/swiftp/' instead ) Just, setup username & password. After tapping Start button, your device will become FTP server & the app will give you URL (handy if you don't know device IP & FTP URL format). On PC, you can use any FTP client software to access file system of device. Windows Explorer works, too. You can also create a network drive in computer. For example, you can visit this in address bar of Windows Explorer:

ftp://192.168.1.10:2121/

Here, 192.168.1.10 is IP of Android device & 2121 is port on which FTP server is working (you can configure port in app, too). Visiting it will ask you username & password. Enter what you specified in app. Done!

The app will not give you URL if device is not connected to PC using external Wi-Fi router, but it works in those situation too. If you've made device Wi-fi Hotspot & connected PC with Ad-hoc network, then you'll have to find device IP by yourself (ipconfig /all command is fine with Windows). It really works as long as you've IP of device (I've tested it by myself). Just, use this IP in FTP URL & visit it.

The Interesting Thing: As FTP is universal standard, this method will work with any PC, Mac. That's not true for Samba file share, AirDroid etc.

Over Bluetooth
There's no need of external app. If Bluetooth software on PC supports OBEX FTP, you can do this directly. If you've issues with stock bluetooth services, install Bluetooth File Transfer app. It supports both OBEX FTP and OPP. Means, you can do both-side file system access: device to PC & PC to device!

Not exactly "accessing the filesystem", but you could try AirDroid. It has a really nice file manager.

There are many ways, you can use ADB over wifi, run an FTP or SSH server on the phone, or use Dropbox.

The easiest to set up is Dropbox, but it's fairly limited which files and you can synchronize. The best solution would be installing an FTP/SSH server on the phone, but setting up those might be difficult.