What is Faster Than Nyquist signaling?
Harry Nyquist made so many contributions that it's easy to get confused.

Related to sampling, Nyquist proved that a signal $s(t)$ bandlimited to $B$ Hz can be reconstructed from samples taken at a rate larger than $2B$. This is unrelated to Faster than Nyquist (FTN) signaling, though.

Related to communications, Nyquist also showed that the maximum  ISI-free pulse rate in a channel of bandwidth $B$ is $2B$ pulses per second, using a technique known as "orthogonal signaling".

Note that Nyquist didn't say that communication at a rate faster than $2B$ was impossible; he just said that you will have to handle the ISI somehow. For a long time, people didn't know how to do that, so $2B$ became engrained as a sort of actual bound.

One exception is partial response signaling, which introduces ISI in a controlled manner. However, in this case the main purpose is spectral shaping more than an increase in data rate.

In the 1970s, Mazo published a paper showing that, in some cases, the ISI can not only be handled, but it actually doesn't detract from a system's error rate performance. Start from an orthogonal signaling system operating at pulse rate $R=1/T$; that is, find an orthonormal set of pulses $p(t-kT)$ (for integers $k,l,m$) such that
$$
p(t-lT)p(t-mT) = \begin{cases}
1,\text{when $l=m$} \\
0,\text{otherwise.}
\end{cases}
$$
Now let's say that you want to transmit the data sequence $\lbrace a_0, a_1, a_2, \ldots, a_M \rbrace$. You can do that by transmitting
$$
s(t) = \sum_{k=0}^M a_k p(t-kT)
$$
and the receiver (ignoring noise for simplicity) can recover any $a_k$ by calculating
$$
a_k = \int_{-\infty}^{\infty} s(t) p(t-kT)\, dt.
$$

Mazo's proposal was to convert this orthogonal-signaling system to an FTN system by introducing a time-compression factor $0 &lt; \tau &lt; 1$, so that the transmitted signal becomes
$$
s(t) = \sum_{k=0}^M a_k p(t-k \tau T)
$$
Note that the bandwidth of $s(t)$ is still $B$, but now we transmit pulses at rate $1/(\tau T) &gt; R$. Mazo's main result was to prove that, for certain pulses $p(t)$ and certain time-compression factors $\tau$, the bit-error rate performance of the accelerated system is the same as the orthogonal system.

At the time, the problem was that handling the ISI was impractical. The ISI acquires a certain structure that can be represented as a trellis, but in any non-trivial case the trellises were huge and well beyond the computational capabilities available.

It is only recently that some new techniques, along with faster computers, have made FTN somewhat practical. I recommend this tutorial paper:

J. B. Anderson, F. Rusek, and V. wall, “Faster-Than- Nyquist Signaling,” Proceedings of the IEEE, vol. 101, no. 8, pp. 1817–1830, Aug. 2013.

Mazo's original paper is:

J. E. Mazo, “Faster-Than-Nyquist Signaling,” Bell System Technical Journal, vol. 54, no. 8, 1975.