This thesis is concerned with developing new methods for performing weak gravita-tional lensing with the aim of addressing specific systematic effects in weak lensingsurveys.The first of these effects is the multiplicative biases which arise as a result ofisotropic smearing. This smearing may be due to atmospheric seeing or an instrumentalPSF. Isotropic smearing circularizes a galaxy image and leads to a systematic under-estimate of the modulus of the observed ellipticity. The orientation of the observedgalaxy is, however, unaffected. We exploit this property by formulating a weak lens-ing shear estimator that requires measurements of galaxy position angles only, therebyavoiding the contribution from this systematic. We demonstrate the method on simula-tions and the CFHTLenS data by reconstructing convergence maps and comparing theresults with the standard full ellipticity based approach. We show that the differencebetween the reconstructed maps for the two approaches is consistent with noise in allof the tests performed. We then apply the technique to the GREAT3 challenge data us-ing three distinct methods to measure the position angles of the galaxies. For all threemethods, we find that the position angle-only approach yields shear estimates with aperformance comparable with current well established shape based techniques.The second effect addressed arises from the intrinsic alignment of the source galax-ies. This alignment mimics a shear signal, and hence biases estimates of the shear. Tomitigate this effect, we develop three shear estimators that include polarization in-formation from radio observations as a tracer of a galaxy’s intrinsic orientation. Inaddition to the shear estimator, we also develop estimators for the intrinsic alignmentsignal. We test these estimators by successfully reconstructing the shear and intrinsicalignment auto and cross-power spectra across three overlapping redshift bins.