Commissioned by the mental health campaign See Me, the YouGov poll of 1,004 adults found 29% reported having had problems themselves.
It also suggested 39% said a family member had been affected.
The research found 35% of respondents believed they or a family member had experienced stigma or discrimination because of their mental health.
See Me Director Calum Irving said: "We all have mental health, it can be up or it can be down, any of us could struggle at any point.
"To treat someone differently because they are going through a tough time isn't fair. However, we know that stigma isn't always intentional. People often don't speak about mental health because they are worried they will say the wrong thing or could make it worse.
"But if no one speaks about mental health, then people won't feel comfortable asking for help when they need it."
Mr Irving added: "A good new year's resolutions for all of us could be to open up a conversation on mental health. You can do that with a simple, 'are you okay?' You don't need to have all the answers, just talking to someone about how they feel can help."
New research has indicated nearly 30% of Scots say they have experienced mental health problems.