Sampling is very important and widely used in various fields. The fundamental theorem in sampling is Shannon’s sampling theorem (see [1]), which deals with band-limited functions that satisfy the Nyquist condition. Two cases are given in Natterer (2001): one is the sampling theorem when the Nyquist condition is over-satisfied; another is the sampling error and remarks when the Nyquist condition is not satisfied. Efforts have been made to generalize Shannon’s sampling theorem in the Clifford algebra setting. In this thesis, we attempt to present the sampling results of the two cases above in Clifford analysis. All of these conclusions are based on the generalized sinc function in [3] and the corresponding PaleyWiener Theorem (see [4]) in R n 1 . The sampling theorem when the Nyquist condition is over-satisfied is firstly discussed in this thesis. We generalize the sampling theorem from n-dimension to (n+1)-real variables in Clifford analysis. The significance of this theorem is the formula in it converges much faster than the sinc series. Secondly, we research on the sampling error formula in Clifford analysis when the Nyquist condition is not satisfied. Remarks are also included.