Laser absorption spectroscopy as a powerful tool for detecting trace gases has been widely used in the monitoring of atmospheric greenhouse gases, pollutions, and respiration processes, including human breath analysis. The detection is based on the light absorption when it propagates through a medium. Most inorganic and organic molecules have characteristic absorption lines in the mid-infrared (mid-IR), which correspond to fundamental vibrational modes and in the near-IR (first overtones) presenting their absorption fingerprints. Here, we summarize the recent developments of the three techniques, namely, wavelength modulation spectroscopy (WMS), cavity ringdown spectroscopy (CRDS), and frequency comb spectroscopy (FCS), and describe their main features as well as possible applications, illustrated by recent experimental results. Emphasis is made on methane detection as applied to breath analysis and atmospheric monitoring. For the WMS technique, we consider local measurements with a multipass cell and also kilometer long open-path configurations for the near-IR and mid-IR spectral regions. The results of measurements of methane in exhaled breath with the CRDS technique in the near-IR are presented for a group of subjects of different ages. We consider various schemes of the FCS that enable fast broadband detection, including direct spectroscopy, dual FCS, and Vernier FCS, and review numerous applications of this approach that revolutionized the field of absorption spectroscopy. The current trends and possible future developments and applications are also discussed.