The reconfigurable intelligent surface (RIS) is a promising technology to provide smart radio environment. In contrast to the well-studied patch-array-based RISs, this work focuses on the metasurface-based RISs and simultaneously transmitting and reflecting (STAR)-RISs where the elements have millimeter or even molecular sizes. For these meticulous metasurface structures, near-field effects are dominant and a continuous electric current distribution should be adopted for capturing their electromagnetic response instead of discrete phase-shift matrices. Exploiting the electric current distribution, a Green's function method based channel model is proposed. Based on the proposed model, performance analysis is carried out for RISs and STAR-RISs. 1) For the transmitting/reflecting-only RIS-aided single-user scenario, closed-formed expressions for the near-field/far-field boundary and the end-to-end channel gain are derived. Then, degrees-of-freedom (DoFs) and the power scaling laws are obtained. It is proved that the near-field channel exhibits higher DoFs than the far-field channel. It is also confirmed that when communication distance increases beyond the field boundary, the near-field power scaling law degrades to the well-known far-field result. 2) For the STAR-RIS-aided multi-user scenario, three practical STAR-RIS configuration strategies are proposed, namely power splitting (PS), selective element grouping (SEG), and random element grouping (REG) strategies. The channel gains for users are derived within both the pure near-field regime and the hybrid near-field and far-field regime. Finally, numerical results confirm that: 1) for metasurface-based RISs, the field boundary depends on the sizes of both the RIS and the receiver, 2) the received power scales quadratically with the number of elements within the far-field regime and scales linearly within the near-field regime.