The internal membranes of eukaryotic cells are all twists and bends characterized by high curvature. During recent years it has become clear that specific proteins sustain these curvatures while others simply recognize membrane shape and use it as "molecular information" to organize cellular processes in space and time. Here we discuss this new important recognition process termed membrane curvature sensing (MCS). First, we review a new fluorescence-based experimental method that allows characterization of MCS using measurements on single vesicles and compare it to sensing assays that use bulk/ensemble liposome samples of different mean diameter. Next, we describe two different MCS protein motifs (amphipathic helices and BAR domains) and suggest that in both cases curvature sensitive membrane binding results from asymmetric insertion of hydrophobic amino acids in the lipid membrane. This mechanism can be extended to include the insertion of alkyl chain in the lipid membrane and consequently palmitoylated and myristoylated proteins are predicted to display similar curvature sensitive binding. Surprisingly, in all the aforementioned cases, MCS is predominantly mediated by a higher density of binding sites on curved membranes instead of higher affinity as assumed so far. Finally, we integrate these new insights into the debate about which motifs are involved in sensing versus induction of membrane curvature and what role MCS proteins may play in biology.