We report on 0.3-10 keV X-ray observations by the Chandra X-ray Observatory of the fields of 22 sources that were discovered as hard X-ray (20-100 keV) sources by the INTEGRAL satellite (IGR sources). The purpose of the Chandra observations is to localize the sources and to measure their soft X-ray spectra in order to determine the nature of the sources. We find very likely Chandra counterparts for 18 of the 22 sources. We discuss the implications for each source, considering previous results and new optical or IR identifications, and we identify or suggest identifications for the nature of 16 of the sources. Two of the sources, IGR J14003-6326 and IGR J17448-3232, are extended on arcminute scales. We identify the former as a pulsar wind nebula (PWN) with a surrounding supernova remnant (SNR) and the latter as a SNR. In the group of 242 IGR sources, there is only one other source that has previously been identified as a SNR. We confirm a previous identification of IGR J14331-6112 as an High-Mass X-ray Binary (HMXB), and we suggest that IGR J17404-3655, IGR J16287-5021, IGR J17354-3255, IGR J17507-2647, IGR J17586-2129, and IGR J13186-6257 are candidate HMXBs. Our results indicate or confirm that IGR J19267+1325, IGR J18173-2509, and IGR J18308-1232 are Cataclysmic Variables (CVs), and we suggest that IGR J15529-5029 may also be a CV. We confirm that IGR J14471-6414 is an Active Galactic Nucleus (AGN), and we also suggest that IGR J19443+2117 and IGR J18485-0047 may be AGN. Finally, we found Chandra counterparts for IGR J11098-6457 and IGR J18134-1636, but more information is required to determine the nature of these two sources.