Since their discovery by Bednorz and Müller (1986), high-temperature cuprate superconductors have been the subject of intense experimental research and theoretical work. Despite this large-scale effort, agreement on the mechanism of high-T c has not been reached. Many theories make their strongest predictions for underdoped superconductors with very low superfluid density n s /m * . For this dissertation I implemented a scanning Hall probe microscope and used it to study magnetic vortices in newly available single crystals of very underdoped YBa 2 Cu 3 O 6+x (Liang et al. 1998. These studies have disproved a promising theory of spin-charge separation, measured the apparent vortex size (an upper bound on the penetration depth λ ab ), and revealed an intriguing phenomenon of "split" vortices.Scanning Hall probe microscopy is a non-invasive and direct method for magnetic field imaging. It is one of the few techniques capable of submicron spatial resolution coupled with sub-Φ 0 (flux quantum) sensitivity, and it operates over a wide temperature range. Chapter 2 introduces the variable temperature scanning microscope and discusses the scanning Hall probe set-up and scanner characterizations. Chapter 3 details my fabrication of submicron GaAs/AlGaAs Hall probes and discusses noise studies for a range of probe sizes, which suggest that sub-100 nm probes could be made without compromising flux sensitivity.The subsequent chapters detail scanning Hall probe (and SQUID) microscopy studies of very underdoped YBa 2 Cu 3 O 6+x crystals with T c ≤ 15 K. Chapter 4 describes two experimental tests for visons, essential excitations of a spin-charge separation theory proposed by Fisher (2000, 2001b). We searched for predicted In regards to the Hall probes, I thank David Kisker (formerly at IBM), and Hadas Shtrikman at Weizmann, for growing the GaAs/AlGaAs 2DEG wafers on which the probes were made. The Marcus group gave me advice on GaAs processes early on. In my attempted measurements of the penetration depth from vortex images, I thank the following people for helpful discussions with us: