A vigesimal (/vɪˈdʒɛsɪməl/ vij-ESS-im-əl) or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). Vigesimal is derived from the Latin adjective vicesimus, meaning 'twentieth'.

In a vigesimal place system, twenty individual numerals (or digit symbols) are used, ten more than in the decimal system. One modern method of finding the extra needed symbols is to write ten as the letter A, or A₂₀ , where the ₂₀ means base 20, to write nineteen as J₂₀, and the numbers between with the corresponding letters of the alphabet. This is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters "A–F". Another less common method skips over the letter "I", in order to avoid confusion between I₂₀ as eighteen and one, so that the number eighteen is written as J₂₀, and nineteen is written as K₂₀. The number twenty is written as 10₂₀.

According to this notation:

In the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example, 10 means ten, 20 means twenty. Numbers in vigesimal notation use the convention that I means eighteen and J means nineteen.

As 20 is divisible by two and five and is adjacent to 21, the product of three and seven, thus covering the first four prime numbers, many vigesimal fractions have simple representations, whether terminating or recurring (although thirds are more complicated than in decimal, repeating two digits instead of one). In decimal, dividing by three twice (ninths) only gives one digit periods (⁠1/9⁠ = 0.1111.... for instance) because 9 is the number below ten. 21, however, the number adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods. As 20 has the same prime factors as 10 (two and five), a fraction will terminate in decimal if and only if it terminates in vigesimal.

The prime factorization of twenty is 2² × 5, so it is not a perfect power. However, its squarefree part, 5, is congruent to 1 (mod 4). Thus, according to Artin's conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37.395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a given set of bases found that, of the first 15,456 primes, ~39.344% are cyclic in vigesimal.

Many cultures that use a vigesimal system count in fives to twenty, then count twenties similarly. Such a system is referred to as quinary-vigesimal by linguists. Examples include Greenlandic, Iñupiaq, Kaktovik, Maya, Nunivak Cupʼig, and Yupʼik numerals.

Vigesimal systems are common in Africa, for example in Yoruba. While the Yoruba number system may be regarded as a vigesimal system, it is complex.^{[further explanation needed]}