Spectrum sensing research has mostly been focusing on narrowband access, and not until recently have researchers started looking at wideband spectrum. Broadly speaking, wideband spectrum sensing approaches can be categorized into two classes: Nyquist-rate and sub-Nyquist-rate sampling approaches. Nyquist-rate approaches have major practical issues that question their suitability for realtime applications; this is mainly because their high-rate sampling requirement calls for complex hardware and signal processing algorithms that incur significant delays. Sub-Nyquist-rate approaches, on the other hand, are more appealing due to their less stringent sampling-rate requirement. Although various concepts have been investigated to ensure sub-Nyquist rates, compressive sampling theory is definitely one concept that has attracted so much interest. This paper explains and illustrates how compressive sampling has been leveraged to improve wideband spectrum sensing by enabling spectrum occupancy recovery with sub-Nyquist sampling rates. The paper also introduces new ideas with great potential for further wideband spectrum sensing enhancements, and identifies key future research challenges and directions that remain to be investigated.