This paper introduces a new class of commutative algebras called *cluster algebras*. Cluster algebras are defined for any positive integer \( n \) and are equipped with a distinguished family of generators called *cluster variables*. These variables are the union of \( n \)-subsets called *clusters*, which satisfy an *exchange property*: any element in a cluster can be replaced by another element related by a *binomial exchange relation*. The paper provides examples of cluster algebras, including the coordinate rings of semisimple groups and Grassmannians. It also discusses the *Laurent phenomenon*, which states that any cluster variable can be expressed as a Laurent polynomial in any given cluster. The authors conjecture that these polynomials have nonnegative integer coefficients and explore the connections between cluster algebras and total positivity, dual canonical bases, and Kac-Moody algebras. The paper is the first in a series that aims to develop the theory of cluster algebras.This paper introduces a new class of commutative algebras called *cluster algebras*. Cluster algebras are defined for any positive integer \( n \) and are equipped with a distinguished family of generators called *cluster variables*. These variables are the union of \( n \)-subsets called *clusters*, which satisfy an *exchange property*: any element in a cluster can be replaced by another element related by a *binomial exchange relation*. The paper provides examples of cluster algebras, including the coordinate rings of semisimple groups and Grassmannians. It also discusses the *Laurent phenomenon*, which states that any cluster variable can be expressed as a Laurent polynomial in any given cluster. The authors conjecture that these polynomials have nonnegative integer coefficients and explore the connections between cluster algebras and total positivity, dual canonical bases, and Kac-Moody algebras. The paper is the first in a series that aims to develop the theory of cluster algebras.