The study of algorithms, computations, and computability offers a major contact point between mathematics, technology, and philosophy. This chapter begins with a brief history of computations and the technical means used to support them. Summary accounts are given of two scholarly developments that provided much of the intellectual background for modern computation: attempts to express all reasoning as mathematics and attempts to reduce all of mathematics to simple, rule-bound symbol manipulation. This is followed by a discussion of the Turing machine, including a detailed explanation of why it can be said to cover all systems of rule-bound symbol manipulation. The universal Turing machine and its philosophical implications are also discussed. A two-dimensional classificatory scheme is offered for proposed constructions of computing devices with stronger computing powers than a Turing machine. This categorization serves to highlight the weaknesses of current proposals for such devices. In conclusion, it is emphasized that computation has to be understood as an intentional input-output process with high demands on reliability and lucidity. The study of computations and algorithms has much to learn from other studies of intentional human action, not least in the philosophy of technology.