I have gone into Composer and specified the page background to be transparent. It still comes out white. Is there a solution?
Steps for QGIS 3.6
A few things have been moved and renamed from QGIS 2.x (see below), this is how it works in 3.x:
Open a new Print Layout (formerly "Composer"): Menu Project - New Print Layout
Right click somewhere on the white empty page, click "Page Properties". In the right sidebar the "Item Properties" tab will be activated, modify the background colour opacity to 0.
For each map item then select it, and in the right sidebar simply untick "Background".
Steps for QGIS 2.4
First make the page background transparent in the sidebar under Composition - Page Settings - Page Background
Then for each map or item go to Item properties - Background and turn that off.
if you are trying to do this in QGIS 3.0, right click on the layout whitespace and select Page Properties. The setting to change the background is at the bottom of this dialog box.
Then you can export to a file format that supports transparency, i.e. PNG, via Composer - Export as image