This paper focuses on determining the opinions of primary school third-grade pupils about the conceptual meaning and use of the numbers "0" and "1". The current study employed the case study design, which is one of the qualitative research methods. In the selection of the sample, the criterion sampling method, one of the purposive sampling methods, was used. The study was conducted with the participation of a total of 114 third-grade pupils (58 girls and 56 boys) attending two state primary schools located in the city of Ankara in the fall term of the 2019-2020 school year. As the data collection tool, a semi-structured interview form developed by the researchers in relation to the concepts of "0" and "1" was used. In the analysis of the data obtained in relation to the concepts of "0" and "1", content analysis was used. The primary school third-grade pupils made sense of the number "0" in association with the categories of four operations, ineffective element, absorbing element, number, natural number, meaningless, valueless, absence, even number, beginning, and letter. The categories in which most thoughts about the meaning of the number "1" are gathered are number, ineffective element, and uniqueness. It was determined that the pupils are more successful in the use of the numbers "0" and "1" in the addition and subtraction operations than in the multiplication and division operations.