This talk will be an introduction to a developing area of analysis called ‘non-commutative function theory’. The guiding theme is how one makes sense of using matrices and linear operators as arguments of functions, with particular focus on how this can be made to work when the arguments are d-tuples of non-commuting operators. Interesting parts of the structure will be highlighted along the way, with particular effort made to indicate how the non-commutative theory contrasts with well-known results in one variable and several commuting variables. This talk is intended to be expository and should be accessible to graduate students.