In this thesis we present the first experimental demonstration of phase-controllable Josephson junctions that carry long range spin-triplet supercurrent. These junctions exhibit ground-state phase shifts of either 0 or $\pi$ and are of considerable interest for the development of random access memory for energy efficient superconducting computers. We demonstrate a scheme by which spin-triplet supercurrent in the junctions is generated through the ferromagnetic proximity effect using three magnetic layers with noncolinear magnetizations. The central layer is a synthetic antiferromagnet with magnetization perpendicular to the plane, while the other two ferromagnetic layers have in-plane magnetization. First, we establish that the supercurrent in these junctions is spin-triplet in nature by observing the characteristic slow decay of the critical current versus the central layer thickness when compared to other junctions that do not have the in-plane layers and carry only spin-singlet supercurrent. The phase state of the junctions is revealed by measuring the interference between two such Josephson junctions in a Superconducting QUantum Interference Device (SQUID) loop. By switching the magnetization of one of the layers by 180$^\circ$ without disturbing the other two layers, we show that the phase state of the Josephson junctions can be controllably switched between 0 and $\pi$ over a thousand times without error, opening possibilities for their use in superconducting memory.We also show that there are easier ways to make a phase-controllable cryogenic memory device using spin-singlet supercurrent. We discuss how Josephson junctions containing only two magnetic layers of appropriate thickness arranged into a spin-valve configuration exhibit controllable 0-$\pi$ switching, first demonstrated by the Birge group at Michigan State University in 2016 using a similar SQUID measurement scheme. I describe the main contributions I made as a part of that effort, in particular the development of a general asymmetric SQUID fitting program that provided the unambiguous proof that the devices switched between the 0 and $\pi$ phase states. We also discuss a number of material studies that served as stepping stones toward the development and improvement of both of the previously mentioned phase-controllable memory demonstrations. We use primarily Fraunhofer physics and SQUID magnetometry to characterize the magnetic and superconducting properties of Josephson junctions containing the ferromagnets: Ni, Ni$_{81}$Fe$_{19}$, Ni$_{65}$Fe$_{15}$Co$_{20}$, Pd$_{97}$Fe$_{3}$, and multilayers of Pd/Co. We examine the relative advantages and disadvantages that each of these materials offer to the development of future superconducting memory devices and compare the strengths and weaknesses of the two phase-control memory schemes.
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