In most developed countries, yes, tap water is generally very safe to drink. Public water systems are strictly regulated by government agencies (like the EPA in the U.S.), which set legal limits for over 90 contaminants. Water providers are required to test their water frequently and provide an annual Water Quality Report (or Consumer Confidence Report) to their customers.
However, safety can be influenced by the age of your home's plumbing (which could introduce lead or copper) or very localised issues. If you have specific concerns, you can request a copy of your local report or use a certified home test kit. For an extra layer of certainty, many people choose to use a point-of-use filter, such as a pitcher filter or faucet attachment.