I History.- 1. Introduction.- 2. The beginnings.- 3. Finitely presented groups.- 4. More history.- 5. Higman's marvellous theorem.- 6. Varieties of groups.- 7. Small Cancellation Theory.- II The Weak Burnside Problem.- 1. Introduction.- 2. The Grigorchuk-Gupta-Sidki groups.- 3. An application to associative algebras.- III Free groups, the calculus of presentations and the method of Reidemeister and Schreier.- 1. Frobenius' representation.- 2. Semidirect products.- 3. Subgroups of free groups are free.- 4. The calculus of presentations.- 5. The calculus of presentations (continued).- 6. The Reidemeister-Schreier method.- 7. Generalized free products.- IV Recursively presented groups, word problems and some applications of the Reidemeister-Schreier method.- 1. Recursively presented groups.- 2. Some word problems.- 3. Groups with free subgroups.- V Affine algebraic sets and the representative theory of finitely generated groups.- 1. Background.- 2. Some basic algebraic geometry.- 3. More basic algebraic geometry.- 4. Useful notions from topology.- 5. Morphisms.- 6. Dimension.- 7. Representations of the free group of rank two in SL(2,C).- 8. Affine algebraic sets of characters.- VI Generalized free products and HNN extensions.- 1. Applications.- 2. Back to basics.- 3. More applicatone.- 4. Some word, conjugacy and isomorphism problems.- VII Groups acting on trees.- 1. Basic definitions.- 2. Covering space theory.- 3. Graphs of groups.- 4. Trees.- 5. The fundamental group of a graph of groups.- 6. The fundamental group of a graph of groups (continued).- 7. Group actions and graphs of groups.- 8. Universal covers.- 9. The proof of Theorem 2.- 10. Some consequences of Theorem 2 and 3.- 11. The tree of SL2.