You can run all tests in a project by executing dotnet test on the terminal. This is handy if you already have the terminal open, but you can add it to Visual Studio code as well.

If you press Cmd-Shift-P to open the Command Palette and type "test", you can run the Run Test Task command. By default, this doesn't do anything, but you can edit tasks.json to tell it how to run dotnet test for you:

tasks.json

{
  "version": "0.1.0",
  "command": "dotnet",
  "isShellCommand": true,
  "args": [],
  "tasks": [
    {
      "taskName": "build",
      "args": [ ],
      "isBuildCommand": true,
      "showOutput": "silent",
      "problemMatcher": "$msCompile"
    },
    {
      "taskName": "test",
      "args": [ ],
      "isTestCommand": true,
      "showOutput": "always",
      "problemMatcher": "$msCompile"
    }
  ]
}

These two task definitions will link the Run Build Task and Run Test Task commands in Visual Studio Code to dotnet build and dotnet test, respectively.

There is a much easier way to run all tests:

1. Install the .NET Core Test Explorer extension
2. Open a .NET Core test project in VS Code, or set dotnet-test-explorer.testProjectPath to the folder path of .NET Core test project in settings.json
3. In .NET Test Explorer of Explorer view, all the tests will be automatically detected, and you are able to run all tests or a certain test

Similar to @Nate Barbettini's answer, but for .Net Core Standard 2.0 (netcoreapp2.0).

{
    "version": "2.0.0",
    "tasks": [
        {
            "label": "test",
            "command": "dotnet test path/to/test-project.csproj",
            "type": "shell",
            "group": "test",
            "presentation": {
                "reveal": "silent"
            },
            "problemMatcher": "$msCompile"
        }
    ]
}

To build on GraehamF's answer, the configuration required in tasks.json for dotnet 2.0 is different.

{
"version": "2.0.0",
"tasks": [
    {
        ...
    },
    {
        "label": "test",
        "command": "dotnet",
        "type": "shell",
        "group": "test",
        "args": [
            "test",
            "${workspaceFolder}/testprojectfolder/testprojectname.csproj"
        ],
        "presentation": {
            "reveal": "silent"
        },
        "problemMatcher": "$msCompile"
    }
]

I found that when Visual Studio and VS Code are both installed, putting the csproj reference in the command property (as in GraehamF's answer) resulted in Visual Studio being opened rather than the tests being run within VS Code.

(I would have put this in a comment, but I do not have enough reputation points.)

To run tests in Visual Studio (VS) Code, you need to add a tasks.json file to the .vscode directory (if you don't already have it). Then configure your test task as follows:

{
    "version": "2.0.0",
    "tasks": [
        {
            ... // Other tasks
        },
        {
            "label": "test",
            "command": "dotnet",
            "type": "shell",
            "args": [
                "test",
                "${workspaceFolder}/TestProjectName/TestProjectName.csproj"
            ],
            "group": "test",
            "problemMatcher": "$msCompile",
            "presentation": {
                "echo": true,
                "reveal": "always",
                "focus": false,
                "panel": "shared",
                "showReuseMessage": true,
                "clear": false
            }
        }
    ]
}

When you have that saved, run the following command Cmd-Shift-P on Mac or Ctrl-Shift-P on Linux and Windows within the VS Code interface then type Run Test Task, press Enter and select test.

The above configuration should work for the latest Insider and Stable releases of VS Code as at April 2020 (version 1.41).