How to set the default namespace in projects using project.json (.xproj)

AFAIK this can't be done with a project.json. You can do it with an xproj the same way you used to do it with a csproj though. Right click it in Visual Studio, and on the Application tab, change the Default namespace.

With ASP.NET Core 1.0.1, you can set your default namespace in the project.json file as follows:

"tooling": {
   "defaultNamespace": "Your.Name.Space"
}

The yeoman ASP.NET generator will respect this defaultNamespace when generating new classes.

For the new Visual Studio 2017 csproj tooling, you can add the following XML to change your default namespace (up in the top level <PropertyGroup> reference):

<PropertyGroup>
  <Optimize>true</Optimize>
  ...
  <RootNamespace>My.Root.Namespace</RootNamespace>
</PropertyGroup>

This is only necessary if your .csproj filename does not match your intended root namespace for the project.