Alt+Space brings up the system menu for the window you’re working in. This menu contains Size and Move options. Choosing Move option turns the cursor into a directional compass, positioned in the centre of the window’s title bar. You can now move the window around on screen with the arrow keys. Press Enter to finish the move operation, or Escape to snap the window back to where it was previously.
The Size option also turns the cursor into a directional compass, this time positioned in the centre of the current window. Pressing any of the arrow keys turns the cursor into a resize cursor, and moves it to the relevant edge of the window. You can now shift that edge of the window either horzontally or vertically. If you want diagonal resizing from the corner of a window, press the up then the right arrow key (or any other horizontal + vertical combination), and use the directional keys to resize that corner in any direction. As with the Move operation, press Enter to fix the resize, or Escape to restore its previous dimensions.
Handy for when you find yourself stuck without a mouse!
Update (30 Jul):
When using this technique to move or resize a window, each press of an arror key moves or resizes the window by about 10 pixels. If you want to change the window position or size in smaller increments, hold down the Ctrl key at the same time for single-pixel control.