Polypeptide analogs of the invention that include a) a base amino acid sequence at least 80% identical to one of a GLP-1 fragment; and b) amino acid residues attached to the carboxy terminus of the base amino acid sequence, where the analogs have GLP-1-like activity of longer duration than native GLP-1 and/or the GLP-1 receptor has a greater affinity for the analogs than native GLP-1. Other polypeptide analogs of the invention include a) a base amino acid sequence at least 50% identical to a GLP-1 fragment in which the amino acid residue in the base amino acid sequence corresponding to the P'.sub.1 residue of GLP-1 is an amino acid analog having a tetrasubstituted C.sub..beta. carbon; and b) amino acid residues attached to the carboxy terminus of the base amino acid sequence, where the analogs have the properties indicated above. The invention also includes methods of treatment where these analogs are administered.