A new nonstandard-analytical approach to quantum fields is presented, which gives a mathematical foundation for manipulating pointwise-defined quantum fields. In our approach, a field operator ϕ (x) is not a standard operator-valued distribution, but a nonstandard operator-valued function. Then formal expressions containing, e.g., ϕ (x)2 can be understood literally, and shown to be well defined. In the free field cases, we show that the Wightman functions are explicitly calculated with the pointwise field, without any regularization, e.g., Wick product. Our notion of pointwise fields is applied also to the path integral formalisms of scalar fields. We show that some of physicists' naive expressions of Lagrangian path integral formulas can be rigorously justified.