Converting a Uniform Distribution to a Fat-tailed Distribution

This previous SO question regards converting a Uniform distribution to a Normal distribution.

For Monte-Carlo simulations, I have a need not only for Normal (Gaussian), but for some computationally efficient ways to generate samples from "fat-tailed" or heavy-tailed distributions, using a given (64-bit or double) uniform RNG as input. Examples of the distributions needed may include: Log-normal, Pareto, Student-T, Cauchy, et.al.

Use of inverse CDFs is acceptable given computationally efficient means of computing the inverse CDF as needed.

The tag is for a language-independant algorithms, but the implementations needed are for basic procedural programming languages (C, Basic, procedural Swift, Python, et.al.)