When a star dies, it leaves a mark on its surrounding environment. The energy from the supernova explosion forms an expanding shock wave that interacts with interstellar and circumstellar material, creating what we know as a supernova remnant (SNR). If the original star has a mass that is greater than or equal to 8 solar masses, this can also lead to the formation of a rapidly rotating neutron star called a pulsar. As these objects evolve, they interact with the surrounding environment, producing non-thermal and thermal emission. For an SNR, its non-thermal emission arises from a population of relativistic particles being accelerated at the shock front of the SNR, while its thermal emission arises from the shock front heating ejecta and and swept-up interstellar medium to X-ray emitting temperatures. For pulsars, their non-thermal emission arises from relativistic particles being accelerated at the termination shock of a pulsar wind. These particles interact with surrounding magnetic fields and ambient photon fields producing synchrotron and inverse Compton emission which we observe as a pulsar wind nebula (PWN), while its thermal emission arises from the surface of the neutron star. These properties of SNRs and pulsars provide a unique window into studying the acceleration, injection, propagation and interaction of highly energetic particles called cosmic rays with the interstellar medium. In addition, they providing information about the evolution, and dynamics of these objects; properties of the shock fronts; details about the original progenitor star; and the impact that these objects have on their surroundings. The research presented here focuses on analysing the intimate connection between cosmic rays, the non-thermal emission arising from SNRs interacting with molecular clouds, and pulsar wind nebulae; as well as analysing the observational and evolutionary properties of these objects. In this thesis we model the propagation of cosmic rays through the Galaxy in an attempt to characterise a standard cosmic ray background with uncertainties, to reveal the origin of the cosmic ray electron positron anomaly. Furthermore, we analyse the gamma-ray emission from SNRs Kes 79 and MSH 11-61A, which are known to be interacting with molecular clouds, as well as the non-thermal X-ray emission arising from the PWN of PSR J1741-2054. We find that the emission from both SNRs most likely arises from the decay of neutral pions that resulted from the interaction of relativistic ions which are accelerated at the shock-front of a SNR, with ambient material. For PSR J1741-2054, we characterise the properties, minimum magnetic field and minimum energy of the particle population that produces the observed diffuse synchrotron emission that surrounds and trails the pulsar.In addition, we characterise the X-ray emission arising from Kes 79, MSH 11-61A and PSR J1741-2054, in an attempt to shed light on the origin and nature of these objects and their emission. Using X-ray data from XMM-Newton and Suzaku respectively, we probe the temperature, ionisation state, and elemental abundance of the shocked gas of each SNR. This allows us to determine their evolutionary properties, properties of the shock, and mass of the original progenitor; and constrain the density of the X-ray emitting plasma. Using Chandra, we determined the temperature of PSR J1741-2054, as well as characterised its proper motion, velocity, direction of motion, and presence of small scale structure immediately surrounding the pulsar.