An ultrasound sparse array consists of a 1 sparse distribution of elements over a 2-D aperture. Such 2 an array is typically characterized by a limited number 3 of elements, which in most cases is compatible with the 4 channel number of the available scanners. Sparse arrays 5 represent an attractive alternative to full 2-D arrays that 6 may require the control of thousands of elements through 7 expensive application-specific integrated circuits (ASICs). 8 However, their massive use is hindered by two main draw-9 backs: the possible beam profile deterioration, which may 10 worsen the image contrast, and the limited signal-to-noise 11 ratio (SNR), which may result too low for some applica-12 tions. This article reviews the work done for three decades 13 on 2-D ultrasound sparse arrays for medical applications. 14 First, random, optimized, and deterministic design meth-15 ods are reviewed together with their main influencing fac-16 tors. Then, experimental 2-D sparse array implementations 17 based on piezoelectric and capacitive micromachined ultra-18 sonic transducer (CMUT) technologies are presented. Sam-19 ple applications to 3-D (Doppler) imaging, super-resolution 20 imaging, photo-acoustic imaging, and therapy are reported. 21 The final sections discuss the main shortcomings associ-22 ated with the use of sparse arrays, the related countermea-23 sures, and the next steps envisaged in the development of 24 innovative arrays. 25 Index Terms-2-D arrays, 3-D ultrasound imaging, 26 capacitive micromachined ultrasonic transducer (CMUT), 27 genetic algorithm (GA), piezoelectric, simulated annealing 28 (SA), sparse arrays, spiral arrays, transducers. 29 I. INTRODUCTION 30 T HE introduction of 2-D arrays has greatly boosted the 31 development of novel ultrasound imaging methods aimed 32 Manuscript