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8-bit
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Eight-bit CPUs normally use an 8-bit data bus and a 16-bit address bus which means that their address space is limited to 64 KBs. This is not a "natural law", however, so there are exceptions.
The first widely adopted 8-bit microprocessor was the Intel 8080, being used in many hobbyist computers of the late 1970s and early 1980s, often running the CP/M operating system. The Zilog Z80 (compatible with the 8080) and the Motorola 6800 were also used in similar computers.

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Encyclopedia
Eight-bit CPUs normally use an 8-bit data bus and a 16-bit address bus which means that their address space is limited to 64 KBs. This is not a "natural law", however, so there are exceptions.
The first widely adopted 8-bit microprocessor was the Intel 8080, being used in many hobbyist computers of the late 1970s and early 1980s, often running the CP/M operating system. The Zilog Z80 (compatible with the 8080) and the Motorola 6800 were also used in similar computers. The Z80 and the MOS Technology 6502 8-bit CPUs were widely used in home computers and game consoles of the 70s and 80s. Many 8-bit CPUs or microcontrollers are the basis of today's ubiquitous embedded systems.
There are 28 (256) possible values for 8 bits.
Why 8 bits?
4-bit microprocessors were developed in the early 1970s starting with the Intel 4004. Intel swiftly followed with 8-bit processors, and most competitors to Intel started with 8-bits. Performance and memory limitations meant that 4-bit processors fell out of more demanding applications quickly (4-bit processors are still mass-produced however).
List of 8-bit CPUs
A CPU can be classified on the basis of the data it can access in a single operation. An 8-bit processor can access 8 bits of data in a single operation, as opposed to a 16-bit processor, which can access 16 bits of data in a single operation.
Examples of 8-bit processors (very incomplete):
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