I will in­tro­duce the pri­mary ob­jects of com­bi­na­to­r­ial group the­ory: group pre­sen­ta­tions, Cay­ley graphs and disc di­a­grams. I will de­scribe some clas­si­cal de­ci­sion prob­lems and then pre­sent Dehn's al­go­rithm (1912) for the so­lu­tion of the word prob­lem for sur­face groups. The proof of the al­go­rithm's cor­rect­ness in­volves cur­va­ture, no­tions of which con­tinue to be de­vel­oped and ap­plied in group the­ory to this day.