Convolutional　neural　networks　(CNN)　have　achieved　remarkable　results　in　various　computer　vision　and　pattern　recognition　applications.　However,　in　computer　graphics　and　geometry　processing,　the　focus　is　on　non-Euclidean　structured　meshed　surfaces.　Since　CNNs　operate　based　on　Euclidean　domains,　the　fundamental　operations　of　CNNs,　such　as　convolution　and　pooling,　are　not　well　defined　in　non-Euclidean　domains.　To　address　this　issue,　we　propose　a　novel　mesh　representation　named　Heat　Kernel　Mesh　(HKM),　which　utilizes　the　heat　diffusion　on　the　non-Euclidean　domain.　The　HKM　represents　a　meshed　surface　as　a　spatio-temporal　graph　signal,　sampled　on　the　edges　of　the　mesh　at　each　time　interval　with　a　Euclidean-like　structure.　Furthermore,　we　propose　the　Heat　Kernel　Mesh-Based　Convolutional　Neural　Network　(HKMCNN),　where　convolution,　pooling,　and　attention　mechanism　are　designed　based　on　the　property　of　our　representation　and　operate　on　edges.　For　the　fine-grained　classification,　we　propose　distance　Heat　Kernel　Mesh　(dHKM)　that　can　identify　discriminant　features　with　the　HKMCNN　to　represent　a　mesh.　Extensive　experiments　on　mesh　classification　and　segmentation　demonstrate　the　effectiveness　and　efficiency　of　the　proposed　HKMCNN.