Corona
In Corona, progressive rendering goes on infinitely by default. You can use rendering limits to make it stop automatically. Corona will stop/finish the rendering when the first limit specified is reached.
Noise limit: The noise level limit can be used to stop Corona's progressive rendering once the specified noise level is reached. The noise level limit is available only if adaptivity is enabled.
Pass limit: Stop the rendering after a certain amount of passes is reached.
Time limit: Limit the render time. This is a "security setting" to avoid too long render times per frame if there is an issue in your scene and it takes very long to reach e.g. the specified noise or pass limit.
Helio recommends always setting a time limit in your scene and finding a noise and/or pass limit that matches your quality requirements.
Note: A good denoiser can save you a lot of render time.
V-Ray
There are two render modes for V-Ray: Progressive and Bucket Rendering.
You'll be able to learn more about it here.
Each of the following criteria is met first, will signal V-Ray to stop rendering.
Progressive rendering
Main options
Noise threshold: Controls how much noise is allowed in the image.
Setting here 0.0 means the entire image is sampled uniformly until either the Max. subdivs value is reached or the Max render time limit is reached.
Max render time: Signals an end to rendering after a certain amount of time has passed.
You can set it to 0.0 to not stop after a certain time.
Additional options
Min. subdivs – Controls the minimum number of samples that each pixel in the image receives. The actual number of the samples is the square of the subdivs.
Max. subdivs – Controls the maximum number of samples that each pixel in the image receives.
Bucket rendering
Blender
Samples are the noise that appears as your scene is rendering. In the render panel, you define the number of samples, and then Blender stops once it reaches it. The more samples, the clearer, but longer your render is.
Redshift
Currently, you need to specify limits in the scene settings / Cinema4D.