Plasma-based accelerators that impart energy gain as high as several GeV to electrons or positrons within a few centimeters have engendered a new class of diagnostic techniques very different from those used in connection with conventional radio-frequency (RF) accelerators. The need for new diagnostics stems from the micrometer scale and transient, dynamic structure of plasma accelerators, which contrasts with the meter scale and static structure of conventional accelerators. Because of this micrometer source size, plasma-accelerated electron bunches can emerge with smaller normalized transverse emittance (n < 0.1 mm mrad) and shorter duration (τ b ∼ 1 fs) than bunches from RF linacs. We review single-shot diagnostics that determine such small n and τ b non-invasively and with high resolution from wide-bandwdith spectral measurement of electromagnetic radiation the electrons emit: n from x-rays emitted as electrons interact with transverse internal fields of the plasma accelerator or with external optical fields or undulators; τ b from THz to optical coherent transition radiation emitted upon traversing interfaces. The duration of ∼ 1 fs bunches can also be measured by sampling individual cycles of a co-propagating optical pulse or by measuring the associated magnetic field using a transverse probe pulse. Because of their luminal velocity and micrometer size, the evolving structure of plasma accelerators, the key determinant of accelerator performance, is exceptionally challenging to visualize in the laboratory. Here we review a new generation of laboratory diagnostics that yield snapshots, or even movies, of laser-and particle-beam-generated plasma accelerator structures based on their phase modulation or deflection of femtosecond electromagnetic or electron probe pulses. We discuss spatiotemporal resolution limits of these imaging techniques, along with insights into plasma-based acceleration physics that have emerged from analyzing the images, and comparing them to simulated plasma structures. CONTENTS I.Introduction 1 II. Properties of plasma accelerator structures and beams 3 A. General properties of plasma electron accelerators 3 1. Frequency-domain holography 41 2. Longitudinal optical shadowgraphy 45 3. Transverse optical probing 46 4. Electron radiography 48 D. "Movies" of wake evolution 49 1. Multi-shot transverse probes 49 2. Single-shot frequency-domain streak camera 50 3. Single-shot imaging of meter-long wakes 52 E. Scaling of wake probes with plasma density 54 V. Conclusion 55 References 58