What is a constant envelope modulation? The Book "Digital Modulation Techniques" by Fuqin Xiong groups digital modulation schemes into two classes: "constant envelope" and "nonconstant envelope."  However, I was surprised by the inclusion of PSK in the "constant envelope" category.  This includes BPSK and QPSK, which are equivalent to PAM and 4-QAM.  This doesn't seem right to me, since both PAM and 4-QAM will have  transitions through zero, unless infinite bandwidth is assumed.

My question then, is what is the definition of a constant envelope modulation?  Is it simply having all symbol constellation points at a fixed energy?  If so, it seems like constant envelope modulation may result in a waveform that is not constant envelope.
You've pointed out a very important distinction between theory and practice. In theory, as suggested by your book and in Fat32's answer, modulation schemes where all information resides in the phase of the signal, not in the amplitude, are called "constant envelope modulation". However, in practice our systems have finite bandwidth and instantaneous phase jumps cannot be realized. Consequently, in such cases the envelope will have dips where (theoretical) phase jumps occur.

Nevertheless, there are systems which really have a constant envelope. Such systems produce a (complex baseband) signal of the form

$$s(t)=Ae^{j\phi(t)}\tag{1}$$

where $\phi(t)$ is a continuous function of time. Such signals are well suited for extremely non-linear channels (amplifiers), and they are also more spectrally efficient than signals with a (theoretically) discontinuous phase.

Modulation schemes producing a signal as given by $(1)$ are called continuous phase modulation (CPM). Examples of CPM are continuous-phase FSK (CPFSK), and minimum-shift keying (MSK) and its variants (e.g., GMSK)