Many older historic cities have a rich tradition of building-sidewalk interaction, but maybe none so famous as New Orleans.
This iconic scene from the French Quarter shows what happens when our built environment and our public walkways interact and intersect with each other; you get one of the most memorable (and profitable) neighborhoods of any American city. This used to be commonplace all over the country prior to the advent of more regulated city development and zoning laws. It is still possible to create new structures that engage with the 'public way'; planning entities and advocates in fact encourage this as a strong strategy for making safe, useful and profitable urban environments.
Many people see the photo above and remark on how beautiful the balconies are, but that's not quite right - it would be more accurate from a planning standpoint to say the French Quarter is known for its beautiful galleries. Balconies, galleries and stoops are all ways buildings historically transitioned into the public way.
To do this today with a new building is possible, but times have changed - you'll now be required to effectively rent that space from the city, and work with the city to ensure that you're not creating any problems by hanging out over their expensive infrastructure.
Balconies
Figure 1. Balcony.
When you think balconies, think Romeo calling to Juliet. These structures cantilever out from the face of the building over the sidewalk which is owned by the local government. This practice was previously accepted in New Orleans as the Way Things Are, but times have changed and cities are strapped for cash. Old overhangs are generally grandfathered from any taxation, but if you want to add a new one hanging out over the sidewalk, you will need a servitude granted to you from the city. This gives you the right to own and maintain the 'encroachment' as they call it, in exchange for an annual payment. In New Orleans, this amount is determined by the city based on real estate prices in the area.
Galleries
Figure 2. Gallery.
Galleries interact with more than just the air, they drop columns into the public path, and into zones that are often needed for public utilities, and where construction equipment must periodically access those utilities. They are taxed the same way as balconies, but they require more extensive review with public agencies and utilities. Generally, New Orleans wants any columns or footings dropped into the sidewalk to hold 18" clear from the face of a curb. The public utilities will look at what they have underground in the area, and will typically ask for 3' of clearance to either side of buried lines in order to work on them should they need replacement. They will also check the height of your encroachment to make sure they can get any equipment they may need underneath it; 10' clear is a safe bet unless you are covering some very serious infrastructure. Private utilites, such as electrical or cable providers, will need to make sure there are no cables that will need to be moved to avoid a safety issue for people standing on the encroachment.
Stoops
Figure 3. Stoop in Brooklyn.
Being a former New Yorker, I have a love for stoops - they are the front porches of Manhattan and Brooklyn. With less sun protection needed, they create a great social scene for a block that we see paralleled here in New Orleans with porches, the difference being porches can be set up as a more private space, whereas stoops are just out there often right in the public sidewalk, forcing interactions - such as the interaction of waking up the guy in the suit who passed out on your stoop so that you can get out the door to work in the morning. That's a little more acceptable than passing out on someone's porch - that would more likely warrant a 911 call than a helpful nudge.