AutoCAD DWG is a closed proprietary file format owned by AutoDesk, most third party applications cannot open it, other than AutoDesk licensed partners for interoperability purposes. Most Blender addons being free can't allocate the resources or justify paying for licensed importer libraries. Linking to them from open source software is also problematic from the legal point of view, so few open source projects can really open DWG files directly.
There is however a built-in importer for DXF file format, if you can save or export your drawing as text-based DXF file. DXF files are complementary exchange file formats and preserve most (not all) DWG information quite well.
It works fairly well in most cases and should import without major hiccups, though some minor bugs are expected.
Fills and stroke widths will not for the most part be preserved, as there is no such structure in Blender. Blender being a 3D modelling tool has no concepts of fills, hatches or line widths.
As of Blender 2.8+ layers are partially preserved, by default all objects belonging to a CAD Layer are grouped together into a single monolithic Blender object. You can optionally turn merging off
For pre Blender 2.8, layers get translated into groups since Blender's current 20 layer system is too limited to accommodate the potentially unlimited number of CAD layers.
You can activate it from settings under User preferences >Addons search for DXF.
Make sure your drawing has no X-lines or Ray objects since those cause import to fail. Also dynamic blocks and other complex structures will likely fail.