probability – Alternatives to Inverse Transform Sampling

I’m interested in writing an algorithm to sample random numbers from some Probability Distribution Function, $P(x)$. We may assume that the Cumulative Distribution Function for $P(x)$, $C(x)$, is also known.

I know that many algorithms use Inverse Transform Sampling to sample pesudo-random numbers from PDF’s. This method finds a random number $n$ according to some PDF $P(x)$ by solving $n = C^{-1}(U)$, with $U approx text{Uniform}(0,1)$ (see for more).

This method works well for PDF’s with easily invertible CDF’s. Suppose, however, that some CDF is difficult or impossible to invert. What other methods (preferably of equal simplicity, if viable) for sampling (psuedo)-random numbers, according to some PDF are available?