This paper presents an advanced 2-D fitting algorithm, GALFIT (Version 3), designed to study the structural components of galaxies and other astronomical objects in digital images. The algorithm allows for the incorporation of irregular, curved, logarithmic, and power-law spirals, ring, and truncated shapes within traditional parametric functions like the Sérsic, Moffat, King, Ferrer, and others. This flexibility enables the mixing and matching of these new shape features, allowing for realistic-looking galaxy model images. Despite the potential for extreme complexity, key parameters such as the Sérsic index, effective radius, and luminosity remain intuitive and unchanged. The new features are useful for quantifying galaxy asymmetry, low surface brightness tidal features, and realistic decompositions of galaxy subcomponents, including strong rings and spiral arms. The paper also discusses the interpretation of model parameters and their uncertainties, and provides case studies to illustrate the capabilities of the new features. The introduction highlights the challenges and benefits of 2-D versus 1-D analysis, emphasizing the advantages of 2-D techniques in handling complex galaxy shapes and image smearing. The paper concludes by detailing the radial profile functions and azimuthal shape functions available in GALFIT, along with their mathematical forms and free parameters.This paper presents an advanced 2-D fitting algorithm, GALFIT (Version 3), designed to study the structural components of galaxies and other astronomical objects in digital images. The algorithm allows for the incorporation of irregular, curved, logarithmic, and power-law spirals, ring, and truncated shapes within traditional parametric functions like the Sérsic, Moffat, King, Ferrer, and others. This flexibility enables the mixing and matching of these new shape features, allowing for realistic-looking galaxy model images. Despite the potential for extreme complexity, key parameters such as the Sérsic index, effective radius, and luminosity remain intuitive and unchanged. The new features are useful for quantifying galaxy asymmetry, low surface brightness tidal features, and realistic decompositions of galaxy subcomponents, including strong rings and spiral arms. The paper also discusses the interpretation of model parameters and their uncertainties, and provides case studies to illustrate the capabilities of the new features. The introduction highlights the challenges and benefits of 2-D versus 1-D analysis, emphasizing the advantages of 2-D techniques in handling complex galaxy shapes and image smearing. The paper concludes by detailing the radial profile functions and azimuthal shape functions available in GALFIT, along with their mathematical forms and free parameters.