Due to its significance in astrobiology, assessing the amount and state of liquid water present on Mars today has become one of the drivers of its exploration. Subglacial water was identified by the Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) aboard the European Space Agency spacecraft Mars Express through the analysis of echoes, coming from a depth of about 1.5 km, which were stronger than surface echoes. The cause of this anomalous characteristic is the high relative permittivity of water-bearing materials, resulting in a high reflection coefficient. A determining factor in the occurrence of such strong echoes is the low attenuation of the MARSIS radar pulse in cold water ice, the main constituent of the Martian polar caps. The present analysis clarifies that the conditions causing exceptionally strong subsurface echoes occur solely in the Martian polar caps, and that the detection of subsurface water under a predominantly rocky surface layer using radar sounding will require thorough electromagnetic modeling, complicated by the lack of knowledge of many subsurface physical parameters. Higher-frequency radar sounders such as SHARAD cannot penetrate deep enough to detect basal echoes over the thickest part of the polar caps. Alternative methods such as rover-borne Ground Penetrating Radar and time-domain electromagnetic sounding are not capable of providing global coverage. MARSIS observations over the Martian polar caps have been limited by the need to downlink data before on-board processing, but their number will increase in coming years. The Chinese mission to Mars that is to be launched in 2020, Tianwen-1, will carry a subsurface sounding radar operating at frequencies that are close to those of MARSIS, and the expected signal-to-noise ratio of subsurface detection will likely be sufficient for identifying anomalously bright subsurface reflectors. The search for subsurface water through radar sounding is thus far from being concluded.