This paper discusses statistical inference in factor analysis, focusing on a general probability model. The authors examine mathematical and statistical problems, including whether observed data uniquely determine the model, and address estimation and hypothesis testing. The paper aims to provide a unified exposition of factor analysis from a mathematical statistics perspective, emphasizing statistical inference for this model. It highlights features of model-building and inference that are relevant in other statistical areas, such as latent structure analysis. The paper also presents new results, with proofs in a technical Part II. The model considered is $ X = \Lambda f + U + \mu $, where $ X $ is a vector of observed variables, $ f $ is a vector of latent factors, $ \Lambda $ is a matrix of factor loadings, and $ U $ is a vector of error terms. The model assumes that $ U $ is independent of $ f $, with mean zero and diagonal covariance matrix $ \Sigma $. The paper distinguishes between two types of models: one where $ f $ is a random vector and another where $ f $ is a vector of nonrandom quantities. The paper discusses the existence of the model, identification, determination of the structure, estimation of parameters, hypothesis testing, determination of the number of factors, and other hypothesis tests. The paper also addresses the problem of determining whether a given covariance matrix $ \Psi $ can be expressed as $ \Sigma + \Lambda\Lambda' $, and provides necessary and sufficient conditions for this. It discusses the identification of $ \Sigma $ and $ \Lambda $, and presents theorems that provide conditions for identification. The paper also considers restrictions that eliminate the indeterminacy of rotation, such as triangularity conditions, diagonality conditions, and simple structure conditions. The paper concludes with a discussion of local identification and the implications of these conditions for statistical inference in factor analysis.This paper discusses statistical inference in factor analysis, focusing on a general probability model. The authors examine mathematical and statistical problems, including whether observed data uniquely determine the model, and address estimation and hypothesis testing. The paper aims to provide a unified exposition of factor analysis from a mathematical statistics perspective, emphasizing statistical inference for this model. It highlights features of model-building and inference that are relevant in other statistical areas, such as latent structure analysis. The paper also presents new results, with proofs in a technical Part II. The model considered is $ X = \Lambda f + U + \mu $, where $ X $ is a vector of observed variables, $ f $ is a vector of latent factors, $ \Lambda $ is a matrix of factor loadings, and $ U $ is a vector of error terms. The model assumes that $ U $ is independent of $ f $, with mean zero and diagonal covariance matrix $ \Sigma $. The paper distinguishes between two types of models: one where $ f $ is a random vector and another where $ f $ is a vector of nonrandom quantities. The paper discusses the existence of the model, identification, determination of the structure, estimation of parameters, hypothesis testing, determination of the number of factors, and other hypothesis tests. The paper also addresses the problem of determining whether a given covariance matrix $ \Psi $ can be expressed as $ \Sigma + \Lambda\Lambda' $, and provides necessary and sufficient conditions for this. It discusses the identification of $ \Sigma $ and $ \Lambda $, and presents theorems that provide conditions for identification. The paper also considers restrictions that eliminate the indeterminacy of rotation, such as triangularity conditions, diagonality conditions, and simple structure conditions. The paper concludes with a discussion of local identification and the implications of these conditions for statistical inference in factor analysis.