The IceCube neutrino observatory is a cubic kilometre-sized Cherenkov detector embedded deep within the glacial ice at the South Pole. It was completed in 2011 and has been gathering data in its full configuration ever since. A central, more densely instrumented sub-detector array known as DeepCore, lowers the neutrino energy threshold of the detector down to 10 GeV. This work summarizes IceCube searches for extraterrestrial point-like sources of neutrinos originating from two different processes, the pair annihilation of gravitationally trapped Dark Matter and interactions of Cosmic Rays at their acceleration sites. Non overlapping samples of events dominated by muon neutrino charged current interactions from atmospheric neutrinos were isolated from 1019 days of operation of IceCube-DeepCore. Atmospheric muon contamination was lowered in comparison to previous searches using innovative veto techniques. An unbinned maximum likelihood ratio method was used to look for an excess of GeV neutrinos in the direction of the Sun among atmospheric neutrinos and muons.Sensitivity towards signals from WIMP annihilations in the Sun was further boosted by using event energy information for events contained within DeepCore. No statistically significant excess was found. Constraints were placed on the muon flux from the Sun, which can also be interpreted as constraints on the WIMP-Nucleon scattering cross section. For spin-dependent scattering, these constraints are the strongest ever, and are stronger than previous constraints from IceCube by an order of magnitude for high WIMP masses. The interpretations of these constraints on some theoretical scenarios of DarkMatter are explored. The unbinned maximum likelihood ratio method has also been used to carry out stacking searches for astrophysical neutrino emissions from catalogs of Galactic supernova remnants,starburst galaxies, galaxy clusters, active galactic nuclei within the Greisen Zatsepin Kuzmin horizon as well as the arrival directions of ultra-high-energy cosmic rays. A search was also carried out for periodic neutrino emission from known Galactic binary systems. No statistically significant excess was found. Constraints are placed on the fraction of the contribution from each class of sources towards the IceCube astrophysical flux, as well as on theoretical flux predictions for specific catalogs.