Lower bounds for the size of a complete partial ovoid in a nondegenerate Hermitian surface are obtained. For even characteristic, a sharp bound is obtained and all examples of this size are described. Next, a general construction method for locally hermitian partial ovoids is explained, which leads to interesting small examples. Finally, a conjecture is given for the size of the largest complete strictly partial ovoid. By using partial derivation, several examples of complete strictly partial ovoids of this size are provided.