\section{Introduction} 
From character expression transfer to the study of variability in human anatomy, non-rigid shape correspondence is the cornerstone of numerous applications. 
It is an inescapable step when transferring or comparing information between 3D shapes, from texture to deformation, and segmentation to motion. 
At scale, correspondences across shape collections enable parametric shape models of various types, such as bodies, faces, hands, and organs \cite{bogo_faust_2014,loper2015smpl,romero2017mano,pavlakos2019smplx}. 
These models are indispensable for numerous downstream tasks: animation pipelines~\cite{igarashi2005rigid}, 3D reconstruction systems~\cite{loper2015smpl}, and morphometric analysis of organs~\cite{el_amrani_universal_2024} are all enabled by parametric models whose power comes from accurate pointwise correspondences. 
Developing non-rigid shape correspondence methods is, therefore, of fundamental importance and an active area of ongoing research.

But such an important task does not come without challenges. 
Data scarcity is a hindrance to development in the field, as geometric 3D acquisitions require specialised technologies, modelling is generally done by expert artists, and annotations necessitate extensive manual work, which can be too ambiguous for shape collections with large variations. 
In the rigid case, a correspondence model can be reduced to a simplified parametrisation with six degrees of freedom \cite{besl1992method}, but this does not hold for \textit{non-rigid deformations}, which are often too complex to obtain an analytical formulation \cite{vestner2017efficient}.

Even the widely studied near-isometric problem setting (e.g., humans simulated in different poses) still requires a deep understanding of geometrical structures to, for instance, distinguish intrinsic symmetries \cite{bogo_faust_2014,wang2025symmetry}. 
As structural differences grow, such as between animals of different species, so does the semantic ambiguity in the matching (how can one map the neck of a giraffe to that of a dog?) \cite{zuffi2017small}. 
Moving to real-world acquisitions, issues such as noise, holes, and partiality are common and severely degrade the performance even of the best available methods. 
Such issues continue to motivate scholars to develop robust, widely applicable shape correspondence tools.

Despite the longstanding nature of the shape correspondence problem, recent years have seen continued progress, even in classical settings such as matching two geometrically complete and clean meshes.
However, researchers have also explored new relevant directions such as extreme non-isometry, zero-shot matching, partiality, and topological changes. 
These challenges inspire new paradigms, such as the application of \textit{foundation model} features in 3D geometric contexts. 
Methods for \textit{partial-to-partial} and \textit{multi-shape matching} address more complex correspondence settings beyond the widely considered \textit{full-to-full} pairwise setting. 

Finally, the progress in non-rigid shape correspondence fields owes much to the parallel evolution of benchmarks. 
Early evaluations focused on synthetic near-isometric shape pairs with uniform meshing, while recent ones have shifted attention toward non-isometry, partiality, real-world noise, and irregular meshing, progressively closing the gap between laboratory and real-world settings.
The scope of shape correspondence is as wide as ever, with exciting application domains for interdisciplinary approaches.

\subsection{Scope of this STAR}

This state-of-the-art report is motivated by recent developments in shape correspondence, both in advancing classical foundations and introducing new paradigms.
Our primary goals are to: (i) summarise the main directions of current research in shape correspondence, and (ii) highlight open problems and future avenues of research.
Our review mainly focuses on works published from 2022 until 2025, including earlier works for background and historical context.

Our report primarily focuses on correspondence problems for 3D meshes, which remain the dominant representation in computer graphics. 
Meshes incorporate rich geometric structure into their surface representation, which has inspired numerous innovative methods that leverage the wealth of knowledge in geometry processing and numerical optimisation.
While point clouds lack such structural information, a review of shape correspondence would be incomplete without mentioning several recent influential approaches as well. Registration of neural implicit representations~\cite{park2019deepsdf,mescheder2019occupancy} is out of our scope in this paper.
Moreover, we do not consider rigid correspondence, as the complexity is vastly reduced when considering transformation models with only six degrees of freedom. 
Image-based feature matching is also beyond the scope of this report; we refer the readers to~\cite{ma2021image} for an overview.

\subsection{Related surveys}  

Previous surveys on shape correspondence have taken various perspectives.
\cite{sahillioglu_recent_2020} covers methods published between 2011 and 2019, many of which provide the foundation for approaches discussed in our report.
The more recent survey reviews correspondence methods up to 2021, with an emphasis on techniques applicable for dynamic shape acquisition and reconstruction \cite{deng_survey_2022}.
We build on both, extend the timeline, introduce novel paradigms, and cover directions that were not fully addressed, such as combinatorial methods.
\cite{van2011survey} summarises shape correspondence methods developed before 2011.
An earlier review on functional maps also offers an excellent theoretical primer~\cite{ovsjanikov_computing_2016}, but focuses on pre-2016 methods and thus excludes the deep learning approaches that dominate the recent literature.

\subsection{Structure}
This STAR is organised as follows.
In \cref{sec:fundamentals}, we introduce the mathematical foundations and problem setting of non-rigid 3D shape correspondence.
The recent body of methods is primarily covered in the next three sections:
\cref {sec:spectral} reviews spectral methods, built on the functional map framework; \cref {sec:combinatorial} covers combinatorial approaches that formulate correspondence as a discrete optimisation problem. 
\cref{sec:deformation} focuses on deformation-based methods that align shapes through spatial transformations.

Following the main body of works, we discuss recent emerging trends that expand the scope of correspondence beyond classical settings in~\cref{sec:emerging}.
Suitable datasets and benchmarks, including recently published ones, are summarised in \cref{sec:datasets}. 
Finally, we outline practical applications of shape correspondence in \cref{sec:applications}, highlight open challenges and discuss future research directions in \cref{sec:challenges}.
