Recent progress in X-ray CT is contributing to the advent of new clinical applications. A common challenge for these applications is the need for new image reconstruction methods that meet tight constraints in radiation dose and geometrical limitations in the acquisition. The recent developments in sparse reconstruction methods provide a framework that permits obtaining good quality images from drastically reduced signal-to-noise-ratio and limited-view data. In this work, we present our contributions in this field. For dynamic studies (3D+Time), we explored the possibility of extending the exploitation of sparsity to the temporal dimension: a temporal operator based on modelling motion between consecutive temporal points in gated-CT and based on experimental time curves in contrast-enhanced CT. In these cases, we also exploited sparsity by using a prior image estimated from the complete acquired dataset and assessed the effect on image quality of using different sparsity operators. For limited-view CT, we evaluated total-variation regularization in different simulated limited-data scenarios from a real small animal acquisition with a cone-beam microCT scanner, considering different angular span and number of projections. For other emerging imaging modalities, such as spectral CT, the image reconstruction problem is nonlinear, so we explored new efficient approaches to exploit sparsity for multi-energy CT data. In conclusion, we review our approaches to challenging CT data reconstruction problems and show results that support the feasibility for new clinical applications.