I'd like to start this collection with a favourite of mine:
Even without the surprising twist in the end, it would make a great story.
The Human voice by StoryCorps
Invention of love
Love the technique they used, even the use of shadows, is just enough to tell a moving story.
Heart of stone (music video)
I lived on the moon (music video)
Eye of the storm (music video)
For the end, something older:
Don't answer me
What are your favourite short animations?