TL;DR

On ASP.NET Core, the access to the request context can be done via the new IHttpContextAccessor interface, which provides a HttpContext property with this information. The IHttpContextAccessor is obtained via dependency injection or directly from the service locator. However, it requires an explicit service collection registration, mapping the IHttpContextInterface to the HttpContextAccessor concrete class, with singleton scope.

Not so short version

System.Web

On classical ASP.NET, the current HTTP context, containing both request and response information, can be accessed anywhere via the omnipresent System.Web.HttpContext.Current static property. Internally, this property uses information stored in the CallContext object representing the current call flow. This CallContext is preserved even when the same flow crosses multiple threads, so it can handle async methods.

ASP.NET Web API

On ASP.NET Web API, obtaining the current HTTP context without having to flow it explicitly on every call is typically achieved with the help of the dependency injection container.
For instance, Autofac provides the RegisterHttpRequestMessage extension method on the ContainerBuilder, which allows classes to have HttpRequestMessage constructor dependencies.
This extension method configures a delegating handler that registers the input HttpRequestMessage instance into the current lifetime scope.

ASP.NET Core

ASP.NET Core uses a different approach. The access to the current context is provided via a IHttpContextAccessor service, containing a single HttpContext property with both a getter and a setter. So, instead of directly injecting the context, the solution is based on injecting an accessor to the context.
This apparently superfluous indirection provides one benefit: the accessor can have singleton scope, meaning that it can be injected into singleton components.
Notice that injecting a per HTTP request dependency, such as the request message, directly into another component is only possible if the component has the same lifetime scope.

In the current ASP.NET Core 1.0.0 implementation, the IHttpContextAccessor service is implemented by the HttpContextAccessor concrete class and must be configured as a singleton.
 

public void ConfigureServices(IServiceCollection services)
{
    services.AddMvc();
    services.AddSingleton<IHttpContextAccessor, HttpContextAccessor>();
}

Notice that this registration is not done by default and must be explicitly performed. If not, any IHttpContextAccessor dependency will result in an activation exception.
On the other hand, no additional configuration is need to capture the context at the beginning of each request, because this is automatically done.

The following implementation details shed some light on this behavior:

• Each time a new request starts to be handled, a common IHttpContextFactory reference is used to create the HttpContext. This common reference is obtained by the WebHost during startup and used for all requests.

• The used HttpContextFactory concrete implementation is initialized with an optional IHttpContextAccessor implementation. When available, this accessor is assigned with each created context. This means that if any accessor is registered on the services, then it will automatically be used to set all created contexts.

• How can the same accessor instance hold different contexts, one for each call flow? The answer lies in the HttpContextAccessor concrete implementation and its use of AsyncLocal to store the context separately for each logical call flow. It is this characteristics that allows a singleton scoped accessor to provide request scoped contexts.

To conclude:

• Everywhere the HTTP context is needed, declare an IHttpContextAccessor dependency and use it to fetch the context.

• Don’t forget to explicitly register the IHttpContextAccessor interface on the service collection, mapping it to the concrete HttpContextAccessor type.

• Also, don’t forget to make this registration with singleton scope.