The (non)existence and complexity of algorithms has been a central theme in combinatorial and, later, geometric group theory since their inception, with low dimensional topology providing both motivation and a significant field of application. In this talk I will review some of the milestones in the development of decision problems in group theory, highlighting the geometry behind them. I shall then survey the current state of the art, with an emphasis on applications to geometry and topology and including decision problems for profinite groups.