Geometric morphometrics is a statistical method for analyzing form based on Cartesian landmark coordinates. After separating shape from size, position, and orientation, Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, a mathematical space derived from shape coordinates, is a metric space that can be locally approximated by a Euclidean tangent space. This allows meaningful assessment of shape similarity and developmental/evolutionary trajectory length and direction. Statistical techniques that preserve these properties, such as principal component analysis, multivariate regression, and partial least squares, can be visualized as actual shapes or deformations. The Procrustes distance between a shape and its relabeled reflection measures bilateral asymmetry. Shape space can be extended to form space by adding the natural logarithm of Centroid Size, a size measure uncorrelated with shape. Thin-plate spline interpolation is the standard tool for computing deformation grids and 3D visualizations, and is central to estimating missing landmarks and the semilandmark algorithm, which allows inclusion of outlines and surfaces in morphometric analysis. The powerful visualization tools and large number of shape variables enable a specific exploratory style of analysis, allowing identification and quantification of previously unknown shape features. Geometric morphometrics is based on landmark coordinates, which are defined as points with names and Cartesian coordinates. Landmarks represent biological homology among forms. Coordinate data can be two- or three-dimensional, captured using digitizing tablets, surface scans, or volumetric scans. Traditional morphometric approaches apply statistical techniques to measurements like distances, angles, areas, and volumes. However, landmark coordinates must be separated from size and nuisance parameters before statistical analysis. The geometry of shape and form is discussed, with shape variables independent of overall size and form variables including size. Shape space is a two-dimensional space for triangles, with Kendall shape space resembling a sphere. Procrustes superimposition is a method to estimate shape parameters by translating, scaling, and rotating landmark configurations. Procrustes shape coordinates are used for analysis, with Procrustes distance measuring shape similarity. Form space includes Centroid Size for size-shape analysis. Deformation grids visualize shape differences, with thin-plate spline interpolation used for 2D and 3D data. Semilandmarks are used for smooth curves and surfaces, allowing statistical analysis of biological features. Statistical analysis in geometric morphometrics uses Procrustes shape coordinates, preserving the original geometry and enabling visualization of results as shape deformations. Methods like principal component analysis, multivariate regression, and partial least squares are used to assess shape variation. Asymmetry is analyzed by comparing shapes with their reflections. Missing data is handled by estimating missing shape coordinates using statistical or geometric methods. Free software packages are available for geometric morphometric analysis, including Morpheus, MorphoJ, and Landmark Editor. The field has advanced significantly, with applications in various biological and paleontological studies.Geometric morphometrics is a statistical method for analyzing form based on Cartesian landmark coordinates. After separating shape from size, position, and orientation, Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, a mathematical space derived from shape coordinates, is a metric space that can be locally approximated by a Euclidean tangent space. This allows meaningful assessment of shape similarity and developmental/evolutionary trajectory length and direction. Statistical techniques that preserve these properties, such as principal component analysis, multivariate regression, and partial least squares, can be visualized as actual shapes or deformations. The Procrustes distance between a shape and its relabeled reflection measures bilateral asymmetry. Shape space can be extended to form space by adding the natural logarithm of Centroid Size, a size measure uncorrelated with shape. Thin-plate spline interpolation is the standard tool for computing deformation grids and 3D visualizations, and is central to estimating missing landmarks and the semilandmark algorithm, which allows inclusion of outlines and surfaces in morphometric analysis. The powerful visualization tools and large number of shape variables enable a specific exploratory style of analysis, allowing identification and quantification of previously unknown shape features. Geometric morphometrics is based on landmark coordinates, which are defined as points with names and Cartesian coordinates. Landmarks represent biological homology among forms. Coordinate data can be two- or three-dimensional, captured using digitizing tablets, surface scans, or volumetric scans. Traditional morphometric approaches apply statistical techniques to measurements like distances, angles, areas, and volumes. However, landmark coordinates must be separated from size and nuisance parameters before statistical analysis. The geometry of shape and form is discussed, with shape variables independent of overall size and form variables including size. Shape space is a two-dimensional space for triangles, with Kendall shape space resembling a sphere. Procrustes superimposition is a method to estimate shape parameters by translating, scaling, and rotating landmark configurations. Procrustes shape coordinates are used for analysis, with Procrustes distance measuring shape similarity. Form space includes Centroid Size for size-shape analysis. Deformation grids visualize shape differences, with thin-plate spline interpolation used for 2D and 3D data. Semilandmarks are used for smooth curves and surfaces, allowing statistical analysis of biological features. Statistical analysis in geometric morphometrics uses Procrustes shape coordinates, preserving the original geometry and enabling visualization of results as shape deformations. Methods like principal component analysis, multivariate regression, and partial least squares are used to assess shape variation. Asymmetry is analyzed by comparing shapes with their reflections. Missing data is handled by estimating missing shape coordinates using statistical or geometric methods. Free software packages are available for geometric morphometric analysis, including Morpheus, MorphoJ, and Landmark Editor. The field has advanced significantly, with applications in various biological and paleontological studies.