We study the foundational properties of persistent homotopy groups and develop elementary computational methods for their analysis. Our main theorems are persistent analogues of the Van Kampen, excision, suspension, and Hurewicz theorems. We prove a persistent excision theorem, derive from it a persistent Freudenthal suspension theorem, and obtain a persistent Hurewicz theorem relating the first nonzero persistent homotopy group of a space to its persistent homology. As an application, we compute sublevelset persistent homotopy groups of alkane energy landscapes and show these invariants capture nontrivial loops and higher-dimensional features that comple- ment the information given by persistent homology.