  The operators of fractional calculus come in many different types, which can
be categorised into general classes according to their nature and properties.
We conduct a formal study of the class known as weighted fractional calculus
and its extension to the larger class known as weighted fractional calculus
with respect to functions. These classes contain tempered, Hadamard-type, and
Erd\'elyi--Kober operators as special cases, and in general they can be related
to the classical Riemann--Liouville fractional calculus via conjugation
relations. Considering the corresponding modifications of the Laplace transform
and convolution operations enables differential equations to be solved in the
setting of these general classes of operators.
