Can pets read human emotions?

I believe that pets read your emotion because they seem to come around you when you are sad or unhappy showing their support. I feel like when you are happy, they seem to get more excited. When you get scared, they want to hide with you. I think they know even if they don't have the knowledge to know what those emotions are. What I mean is that they know that we are having those emotions even though they don't know what the emotion is exactly.