We address the problem of designing compact data structures for encoding a Triangulated Irregular Network (TIN). In particular, we study the problem of compressing connectivity, i.e., the information describing the topological structure of the TIN, and we propose two new compression methods which have different purposes. The goal of the first method is to minimize the number of bits needed to encode connectivity information: it encodes each vertex once, and at most two bits of connectivity information for each edge of a TIN; algorithms for coding and decoding the corresponding bitstream are simple and efficient. A practical evaluation shows compression rates of about 4.2 bits per vertex, which are comparable with those achieved by more complex methods. The second method compresses a TIN at progressive levels of detail and it is based on a strategy which iteratively removes a vertex from a TIN according to an error-based criterion. Encoding and decoding algorithms are presented and compared with other approaches to progressive compression. Our method can encode more general types of triangulations, such as those constrained by topographic features, at the cost of a slightly longer bitstream.