In many parts of the States you do, but the strictness of the test varies and the standards are grandfathered.
So, for example, in California (one of the strictest) you must pass a visual inspection, a tailpipe sniffer, and an engine computer scan. There are also many rules about acceptable engine modifications. In other states you may only get an engine computer scan.
As for standards, cars are (rather reasonably) held to the standard they were made to, not the standard of today, and modern cars really are dramatically less polluting than cars from several decades earlier.