As of 3Q2013, there are a few leading choices for 2D CAD software for Linux. In short, if you have a specific need to work natively with DWG (e.g. customer demands DWG, and DXF is not acceptable to them (sigh)), then you can try the free-of-charge but non-libre Draftsight.
If you like libre software, you can consider: QCAD or LibreCAD — then use the free Teigha convertor mentioned below as a first/last step to convert to/from DWG DXF.
I have not tried nanoCAD as nanoCAD is currently free but non-libre.
I didn’t include FreeCAD here as it’s mainly for 3D (and also cannot open DWG at this time).
Draftsight is the “easy” solution that typically just works with most DWG someone sends you — but at the price of being non-libre.
QCAD 3 is a free, GPLv3 solution, and can work with DWG–via a non-libre, non-free DWG package (or use the free, non-libre Teigha DXF – DWG converter)
LibreCAD 2 is a GPLv2 2D CAD solution — but canNOT nativly open DWG files due to the present lack of a suitably GPL’d DWG import/export package — so consider using the free, non-libre Teigha DXF – DWG converter
License: no monetary cost, but NOT libre
Registration: YES: to install AND REregistration at periodic intervals <–annoying
Download: binary 32-bit .deb package
DWG: YES: through R2010 (as of Draftsight V1R3.1)
DXF: YES: through R2010 (as of Draftsight V1R3.1)
Export: several, including: PDF, SVG, EPS, PNG, BMP…
License: GPLv3 (base package only; the optional packages for DWG, etc. are paid license)
Download: Source code, or convenient 64-bit or 32-bit binary packages that works on most Linux distros
DWG: YES: via paid proprietary package through R2013 — 15 minute free trial can apparently be used over and over? (or use free Teigha DXF-DWG converter)
DXF: YES: through R2013
Export: several, including: PDF, SVG,PNG,BMP…
Download: Source code, or convenient PPA for Ubuntu (64-bit and 32-bit)
DWG: NO: but you can use free, non-libre Teigha DXF-DWG converter as a first/last step to/from DWG-DXF
DXF: YES: through R2007
Export: several, including: EPS, SVG,PNG,BMP…
There is a free, non-libre “Teigha File Converter” software http://www.opendesign.com/guestfiles/TeighaFileConverter
to convert from DXF to DWG en masse as a first/last step (compatible through R2013 DWG/DXF).
This OpenDesignAlliance is the same organization where QCad gets its non-free non-libre converter from.