"Rate This App"-link in Google Play store app on the phone

Kotlin solution (In-app review API by Google in 2020):

You can now use In app review API provided by Google out of the box.

First, in your build.gradle(app) file, add following dependencies (full setup can be found here)

dependencies {
    // This dependency is downloaded from the Google’s Maven repository.
    // So, make sure you also include that repository in your project's build.gradle file.
    implementation 'com.google.android.play:core:1.8.0'
    implementation 'com.google.android.play:core-ktx:1.8.1'
}

Create a method and put this code inside:

val manager = ReviewManagerFactory.create(context)
val request = manager.requestReviewFlow()
request.addOnCompleteListener { request ->
    if (request.isSuccessful) {
        // We got the ReviewInfo object
        val reviewInfo = request.result
        val flow = manager.launchReviewFlow(activity, reviewInfo)
        flow.addOnCompleteListener { _ ->
          // The flow has finished. The API does not indicate whether the user
          // reviewed or not, or even whether the review dialog was shown. Thus, no
         // matter the result, we continue our app flow.
        }
    } else {
        // There was some problem, continue regardless of the result.
    }
}

Source

Here is a working and up to date code :)

/*
* Start with rating the app
* Determine if the Play Store is installed on the device
*
* */
public void rateApp()
{
    try
    {
        Intent rateIntent = rateIntentForUrl("market://details");
        startActivity(rateIntent);
    }
    catch (ActivityNotFoundException e)
    {
        Intent rateIntent = rateIntentForUrl("https://play.google.com/store/apps/details");
        startActivity(rateIntent);
    }
}

private Intent rateIntentForUrl(String url)
{
    Intent intent = new Intent(Intent.ACTION_VIEW, Uri.parse(String.format("%s?id=%s", url, getPackageName())));
    int flags = Intent.FLAG_ACTIVITY_NO_HISTORY | Intent.FLAG_ACTIVITY_MULTIPLE_TASK;
    if (Build.VERSION.SDK_INT >= 21)
    {
        flags |= Intent.FLAG_ACTIVITY_NEW_DOCUMENT;
    }
    else
    {
        //noinspection deprecation
        flags |= Intent.FLAG_ACTIVITY_CLEAR_WHEN_TASK_RESET;
    }
    intent.addFlags(flags);
    return intent;
}

Put the code in the Activity you would like to call it from.
When the user clicks a button to rate the app, just call the rateApp() function.

I open the Play Store from my App with the following code:

            val uri: Uri = Uri.parse("market://details?id=$packageName")
            val goToMarket = Intent(Intent.ACTION_VIEW, uri)
            // To count with Play market backstack, After pressing back button, 
            // to taken back to our application, we need to add following flags to intent. 
            goToMarket.addFlags(Intent.FLAG_ACTIVITY_NO_HISTORY or
                    Intent.FLAG_ACTIVITY_NEW_DOCUMENT or
                    Intent.FLAG_ACTIVITY_MULTIPLE_TASK)
            try {
                startActivity(goToMarket)
            } catch (e: ActivityNotFoundException) {
                startActivity(Intent(Intent.ACTION_VIEW,
                        Uri.parse("http://play.google.com/store/apps/details?id=$packageName")))
            }

Option 2: is to use resolveActivity instead of try..catch

if (sendIntent.resolveActivity(getPackageManager()) != null) {
     startActivity(chooser);
} else {
    openUrl();
}

I always use this code:

startActivity(new Intent(Intent.ACTION_VIEW, Uri.parse("market://details?id=PackageName")));