This survey provides a comprehensive overview of Homomorphic Encryption (HE) schemes, focusing on their theory and implementation. HE allows third parties to perform operations on encrypted data without needing to decrypt it first, addressing privacy concerns in cloud services. The survey covers Partially Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which are foundational to Fully Homomorphic Encryption (FHE). FHE, introduced by Craig Gentry in 2009, enables any computable function to be applied to encrypted data. However, FHE schemes have significant practical challenges, such as computational overhead and complexity. The survey discusses various FHE families, including ideal lattice-based, integer-based, and (R)LWE-based schemes, along with their implementations and improvements. It also highlights the contributions of key researchers and the challenges and future directions in HE research. The goal is to provide a clear foundation for researchers and practitioners interested in understanding, applying, and advancing HE systems.This survey provides a comprehensive overview of Homomorphic Encryption (HE) schemes, focusing on their theory and implementation. HE allows third parties to perform operations on encrypted data without needing to decrypt it first, addressing privacy concerns in cloud services. The survey covers Partially Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which are foundational to Fully Homomorphic Encryption (FHE). FHE, introduced by Craig Gentry in 2009, enables any computable function to be applied to encrypted data. However, FHE schemes have significant practical challenges, such as computational overhead and complexity. The survey discusses various FHE families, including ideal lattice-based, integer-based, and (R)LWE-based schemes, along with their implementations and improvements. It also highlights the contributions of key researchers and the challenges and future directions in HE research. The goal is to provide a clear foundation for researchers and practitioners interested in understanding, applying, and advancing HE systems.