cannot find module 'angular2/core'

This is how I fixed the problem in VS2015 :

• Open to the project property window
• Navigate to the TypeScript Build tab
• Select CommonJS as module system as shown in image below

Here is my working settings:

<PropertyGroup Condition="'$(Configuration)' == 'Release'">
  <TypeScriptTarget>ES5</TypeScriptTarget>
  <TypeScriptJSXEmit>None</TypeScriptJSXEmit>
  <TypeScriptCompileOnSaveEnabled>True</TypeScriptCompileOnSaveEnabled>
  <TypeScriptNoImplicitAny>False</TypeScriptNoImplicitAny>
  <TypeScriptModuleKind>System</TypeScriptModuleKind>
  <TypeScriptRemoveComments>False</TypeScriptRemoveComments>
  <TypeScriptOutFile />
  <TypeScriptModuleResolution>NodeJs</TypeScriptModuleResolution>
  <TypeScriptOutDir />
  <TypeScriptGeneratesDeclarations>False</TypeScriptGeneratesDeclarations>
  <TypeScriptNoEmitOnError>True</TypeScriptNoEmitOnError>
  <TypeScriptSourceMap>True</TypeScriptSourceMap>
  <TypeScriptMapRoot />
  <TypeScriptSourceRoot />
  <TypeScriptEmitDecoratorMetadata>True</TypeScriptEmitDecoratorMetadata>
  <TypeScriptExperimentalDecorators>True</TypeScriptExperimentalDecorators>
</PropertyGroup>

Restart Visual Studio after applying this to your web project.

For VS2015, go to Tools -> Options -> Text Editor -> TypeScript -> Project -> General then tick "Automatically compile TypeScript files which are not part of a project" then select "Use CommonJS code generation for modules.."

For ASP.NET 5.0 applications in VS 2015, configuring typescript is a bit challenging.

Until the tooling around typescript improves, you can configure typescript manually:

Step 1: Right-click project, and Unload Project 
Step 2: Right-click the unloaded project, and Edit the .xproj file
Step 3: Add a PropertyGroup node, under the Project node:

  <PropertyGroup Condition="'$(Configuration)' == 'Debug'">
    <TypeScriptTarget>ES5</TypeScriptTarget>
    <TypeScriptJSXEmit>None</TypeScriptJSXEmit>
    <TypeScriptCompileOnSaveEnabled>True</TypeScriptCompileOnSaveEnabled>
    <TypeScriptNoImplicitAny>False</TypeScriptNoImplicitAny>
    <TypeScriptModuleKind>CommonJS</TypeScriptModuleKind>
    <TypeScriptRemoveComments>False</TypeScriptRemoveComments>
    <TypeScriptOutFile />
    <TypeScriptOutDir />
    <TypeScriptGeneratesDeclarations>False</TypeScriptGeneratesDeclarations>
    <TypeScriptNoEmitOnError>True</TypeScriptNoEmitOnError>
    <TypeScriptSourceMap>True</TypeScriptSourceMap>
    <TypeScriptMapRoot />
    <TypeScriptSourceRoot />
    <TypeScriptExperimentalDecorators>True</TypeScriptExperimentalDecorators>
  </PropertyGroup>

Step 4: Right-click the unloaded project and Reload Project
Step 5: Re-build project

If you are still encountering the error where it cannot find the module, exit visual studio and reload the solution.