Phytantriol is an amphiphilic lipid capable of self-assembly. It exhibitsa range of mesophases including a bicontinuous cubic phase, which is stablein excess water. The cubic phase has potential applications in soft matter,such as cubosome carriers for drug delivery, and as a template for creatinghard matter, such as its use as scaffolding templates in electrodeposition. Keyto these applications is ion transport through the mesophases, which is thesubject of this thesis. Ion transport is determined by physical specifics of themesophases such as nanostructure geometry, domain boundaries, and Debyelength. The study of ion transport therefore, can shed light on the mesophasesthemselves.This thesis presents experimental studies of the conductance of phytantriolmesophases and their lyotropic and thermotropic phase transitions. In addition,a method to investigate how conductance through phytantriol is affectedby strain is presented and demonstrated. The conductance of the variousthermotropic and lyotropic mesophases, along with their respective transitions,have been compared by continuous conductance measurements with varyinghydration and temperature. Simultaneous imaging with cross-polarizedmicroscopy enabled the conformation of transitions to and from optically anisotropic phases. Both discontinuous and continuous changes were observedacross transitions, reflecting their structural reconfiguration. Unexpectedly,it is shown that the conductance of the inverse micellar phase can becomecomparable to the lamellar phase. Also unexpected was the observation of abirefringent hysteretic phase not reported in previous literature, based on studiesrestricted to the steady state. While these measurements provide a globalcomparison between mesophases, there are restrictions to interpretation dueto the physical complexity of the system. The remainder of the thesis focuseson the Q224 phase, which is stable in excess water and most relevant to knownpotential applications.Four-probe electrical measurements on a micropore filled with Q224 phaseprovided accurate conductivity values and show that transport is reduced by afactor of 30-40 compared to bulk solution. The effect of temperature and highelectrostatic bias were also investigated. By increasing the temperature, it isshown that a transition to the HII phase can be induced, leading to a switchin resistivity. With large electrostatic bias, if the Q224 phase is applied as anasymmetrical deposit, it is shown that the deposit can be deformed also leadingto a different resistance, demonstrating that the resistance can be electricallyswitched.An experimental set-up was developed in order to investigate the effects ofstress and deformation on the ion transport of the cubic phase. A size variablepore is used in order to apply strain to the mesophase material. Upon changingpore size, the cubic phase exhibits transient behaviour showing positive andnegative piezoresistance. The negative piezoresistance is up to 4 times greaterthan the expected resistance due to changing pore geometry. When thepore is stretched slowly by small increments, the behaviour can be comparedagainst an open pore and the data suggest that the meso-phase conducts morewith stretching. With further development ion transport and a size variablepore could be used to measure the electro-rheological response of mesophasematerials.