Airspace classification

The world's nations claim the right to control their own airspace (i.e. the portion of the atmosphere above their territorial jurisdiction) up to reaching space limit (or between 80 to 100 km -50 to 62 miles- from sea level). For navigational purposes, this airspace is typically subdivided into classes, each with their own flying rules. Airspace classification and their flying rules may vary between countries, although most countries follow either ICAO's classification or a modified version of it. Airspace limits are shown in official charts.

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US airspace classification (image embedded from the Aerospace Power Journal on 27 March 2010) Airspace representation on a navigation chart (image embedded from Pirep on 27 March 2010)

According to ICAO, the national airspace can be classified as pertaining to one of seven classes. Five of these classes fall within the category "controlled airspace", while two of the classes pertain to the category "uncontrolled airspace". Controlled or uncontrolled airspace refers to whether the pilot has to seek permission and obey ATC directions or not, respectively.

Controlled airspace
Controlled airspace describes airspace which such traffic levels that air traffic control (ATC) must provide some form of separation between aircraft.
Class A All operations must be conducted under Instrument Flight Rules (IFR) or Special visual flight rules (SVFR) and are subject to ATC clearance. All flights are separated from each other by ATC. This airspace ranges from 18,000 to FL600. All operation in this airspace must be under instrument flight rules and are subject to air traffic control clearances and instructions (Pilots must file and fly an IFR flight plan). At altitudes between FL180 and FL450, jet routes and high routes will be classified as Class A airspace.
Class B Operations may be conducted under IFR, SVFR, or Visual flight rules (VFR). All aircraft are subject to ATC clearance. All flights are separated from each other by ATC. The rules for this airspace have changed slightly. Pilots must maintain "clear of clouds" with three miles visibility. Pilots must be "cleared" to enter class B airspace, have two-way radio communication and mode C transponders.
Class C Operations may be conducted under IFR, SVFR, or VFR. All flights are subject to ATC clearance. Aircraft operating under IFR and SVFR are separated from each other and from flights operating under VFR. Flights operating under VFR are given traffic information in respect of other VFR flights. There were no changes made to the operating and equipment rules for the Class C airspace. Cloud requirements are 500' below, 1000' above, and 2000' horizontal. Visibility is 3 miles. Pilots must have two-way radio communication and mode C transponders to enter class C airspace.
Class D Operations may be conducted under IFR, SVFR, or VFR. All flights are subject to ATC clearance. Aircraft operating under IFR and SVFR are separated from each other, and are given traffic information in respect of VFR flights. Flights operating under VFR are given traffic information in respect of all other flights. These are control zones (CZ) with operating control towers and airport traffic areas (ATA). A segmented blue line depicts class D airspace on sectional and world aeronautical charts. The ceiling of Class D airspace is 2500 AGL (charted in MSL), and the lateral dimensions are dependent upon the instrument approaches at that airport. Two-way radio communication is required. Cloud and visibility clearance is 3 miles, with 500' below, 1000' above, and 2000' horizontal. Pilots must have two-way radio communication to enter class D airspace.
Class E Operations may be conducted under IFR, SVFR, or VFR. Aircraft operating under IFR and SVFR are separated from each other, and are subject to ATC clearance. Flights under VFR are not subject to ATC clearance. As far as is practical, traffic information is given to all flights in respect of VFR flights. This airspace includes all federal airways, continental control areas, control area extensions, control zones at airports without control towers, transition areas, and area low routes. On the sectional charts, Class E airspace around airports without control towers is depicted with a segmented magenta line. The floor of the Class E airspace is 700 AGL, 1200 AGL, 1500 AGL, or 14,500 AGL, whichever applies. Cloud and visibility clearances are 3 miles, with 500' below, 1000' above, and 2000' horizontal.
Uncontrolled airspace
Uncontrolled airspace describes airspace where an Air Traffic Control (ATC) service is not deemed necessary or cannot be provided for practical reasons.
Class F Operations may be conducted under IFR or VFR. ATC separation will be provided, so far as practical, to aircraft operating under IFR. Traffic Information may be given as far as is practical in respect of other flights. Operations may be conducted under IFR or VFR. ATC separation will be provided, so far as practical, to aircraft operating under IFR. Traffic Information may be given as far as is practical in respect of other flights.
Class G Operations may be conducted under IFR or VFR. ATC separation is not provided. Traffic Information may be given as far as is practical in respect of other flights. The vertical limits for this airspace are 700 AGL, 1200 AGL, 1500 AGL, or 14,500 MSL, whichever applies. Cloud and visibility clearances are 1 mile and clear of clouds.

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Wikipedia - Airspace class
Wikipedia provides additional information on airspace classification, especially regarding some international differences and their rules.

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