Whispering gallery modes (WGMs) in dielectric microresonators with circular symmetry have become the focus of many theoretical and experimental studies due to their exceptional properties such as extremely small mode volumes, very high power density and narrow spectral linewidths. WGMs have found many applications ranging from unique tools to study nonlinear optical phenomena and quantum electrodynamics, to low threshold microlasers as well as very sensitive microsensors. Among the variety of optical microresonators shapes, materials and technologies, optical fiber micro resonators offer additional advantages of simpler and less expensive fabrication. This tutorial will introduce the fundamentals of fiber microresonators, review their fabrication methods, optical coupling techniques and applications for ultrasensitive detection of gases, volatile organic components (VOCs), liquid analytes and other physical parameters. Specific focus will be on the sensors based on polymer-coated microfiber resonators that rely on measurements of refractive index changes due to absorption of gas molecules by the coatings. Several specific examples of the microresonator based gas/VOC sensors and sensors will be presented including analysis and comparison of their performance. Microresonators for sensing of other parameters will also be briefly discussed. In conclusion, most recent developments, future prospects and challenges for the gas/VOC sensors will be discussed including the need for simultaneous detection of multiple gases/VOCs and overcoming the drawbacks of fabrication and readout complexity.