An adhesive composition made from an elastomer and a tackifying resin. The tackifying resin includes a farnesene polymer or copolymer having the following properties: i) less than 10 weight percent of volatile organic compounds; ii) Mn between 300 Da and 1000 Da; iii) Mw between 400 Da and 3000 Da; iv) Mw/Mn between 1.00 and 3.00; v) Tg between −50° C. and 20° C.; and vi) viscosity between 400,000 cP and 1,000,000 cP at 25° C. A method of making the farnesene-based polymer or copolymer includes combining a farnesene monomer and a solvent and optionally adding one or more co-monomers selected from dienes, branched mono-olefins, and vinyl aromatics, to provide a monomer feed, and polymerizing the monomer feed by combining it with a Friedel-Crafts initiator in a vessel. The farnesene-based polymer or copolymer tackifier may be combined with one or more elastomers and one or more other tackifiers to form an adhesive composition.