Curve similarity plays a crucial role in various domains where comparing functional or dynamic shapes is essential, including bioassay analysis, trajectory studies, spectroscopy, medical signal interpretation, and functional genomics. Despite its broad impact, research on curve similarity methods remains fragmented across statistical, computational geometry, and signal processing communities, leading to a lack of unified terminology and systematic comparison. To address this gap, this study adopts a structured literature review methodology, in which relevant studies are identified through a comprehensive search of major academic databases and selected based on predefined inclusion criteria, including peer-reviewed publications focusing on similarity measures for curves and time series. The review systematically examines mathematical and statistical approaches to curve similarity, focusing on their theoretical foundations, statistical properties, and practical applications. The selected methods are categorized into five groups: distance-based, alignment-based, topology-oriented, statistical and hypothesis testing, and learning-based approaches. For each category, key aspects such as mathematical formulation, invariance properties, robustness to sampling variability, and computational characteristics are analyzed. In addition, application domains, method comparisons, and common limitations are discussed, along with available software tools that support curve similarity analysis. By providing a structured and methodologically grounded synthesis, this review assists researchers in selecting appropriate techniques and highlights potential directions for developing more robust and scalable similarity assessment frameworks.