Yes, London has an area actually called The City of London. Technically, this is London proper and what most of us think of as London is this plus all the other boroughs that make up what is referred to as Greater London. The City is the hub for business in London, so you will see many flagship stores throughout the area as well as major company headquarters. This is also where London's financial services core is based.
The City, as it's called by locals (sometimes it's also referred to as the "Square Mile"), is the oldest part of the city, so you can see some of the best architecture, starting with the St. Paul’s Cathedral. If you are looking for something more modern, Barbican Centre features art galleries, live performances, and cinema.
Nearly all of the housing here is in the form of apartments, though there are a few terraced homes for those with a healthy bank account.
Canary Wharf: 3.3 miles / 15-30 minutes by car / 15 minutes by transit / 25 minutes by bike
Southwark: 2.6 miles / 15-35 minutes by car / 30 minutes by transit / 20 minutes by bike
Westminster: 2.6 miles / 10-35 minutes by car / 15-20 minutes by transit / 15-20 minutes by bike
Shoreditch: 1.7 miles / 10-25 minutes by car / 15-20 minutes by transit / 30 minutes by foot / 10-15 minutes by bike
Photo Credits: Thanks to @thebestoflondon_, @human.of.london, @ciudadoflondon, @explorebeautifullondo, @visitthecity, @london_investor, @evievaevita, @jesphine, @alvaruenda, @rodericklonsdale, @annepenthe, @broker_98, @kendall_lisa, @marilouloudis, @andreaminari_ for your great photos of this neighborhood!