This paper presents an introduction to the astrophysics of cosmic rays and diffuse gamma-rays and discusses some of the puzzles that have emerged recently due to more precise data and improved propagation models: the excesses in Galactic diffuse gamma-ray emission, secondary antiprotons and positrons, and the flatter than expected gradient of cosmic rays in the Galaxy. These also involve the dark matter, a challenge to modern physics, through its indirect searches in cosmic rays. Though the final solutions are yet to be found, I discuss some ideas and results obtained mostly with the numerical propagation model GALPROP. A fleet of spacecraft and balloon experiments targeting these specific issues is set to lift off in a few years, imparting a feeling of optimism that a new era of exciting discoveries is just around the corner. A complete and comprehensive discussion of all the recent results is not attempted here due to the space limitations.