Background: Radiation in space poses a threat to humans embarked on missions to the Moon or Mars. Several studies deal with allowable doses, levels of radiation doses in space, and effects of various forms of shielding. The recent shift in emphasis from to 95% confidence intervals adds significantly to the challenge in designing human space missions. Recent reports issued by NASA as well as the Exploration Systems Architecture Study (ESAS) have estimated radiation effects for some mission scenarios. Nevertheless, radiation effects and the effectiveness of shielding remain uncertain. Method: Models and data in the literature are reviewed, and comparisons are made between allowable dose and estimated dose for lunar and Mars missions. Appraisals are made of the feasibility of providing radiation protection for crews in human missions to the Moon and Mars. A number of investigators have prepared point estimates of the doses due to galactic cosmic radiation (GCR) or solar particle events (SPE) for specific locations in space. However, the current NASA trend is to utilize the 95th percentile confidence interval (CI) rather than the point estimate for dose. In cases where the 95% CI has been modeled, the 95% CI dose is typically 3 to 4 times the point estimate. We have therefore multiplied point estimates by ~3.5 to roughly approximate 95% CI estimates, and compared them with allowable doses for various cases: (a) in space, (b) in space behind shields, (c) on the lunar surface behind various shields or within habitats, and (d) on the surface of Mars behind shields or within habitats. Conclusion: For lunar sortie missions, the duration is short enough that GCR creates no serious risks. For lunar outpost missions the probability of encountering an SPE during Solar Maximum in a 6-month rotation is 1% to 10% depending on the assumed energy of the SPE. Even with > 30 g/cm of regolith shielding the 95% CI dose from a major SPE would exceed the 30-day limit. The GCR during Solar Minimum for a 6-month stay on the Moon is marginal against the annual limit, but this can be mitigated somewhat by use of regolith for shielding the habitat. For Mars missions, we conjecture a 400-day round trip transit to and from Mars, and about 560 days on the surface. The GCR 95% CI GCR dose equivalent with 15 g/cm of aluminum shielding during Solar Minimum is about double the allowable annual dose for each leg of the trip to and from Mars. If a major SPE occurred during a transit, the crew would receive a sufficient dose to reduce their life expectancy by more than the 3% limit. The probabilities of encountering a large SPE are ~2.4% for a 4X 1972 SPE and about 20% for a 1X 1972 SPE in a round trip of 400 days during Solar Maximum. On the surface of Mars, the accumulated GCR 95% CI dose over the course of a year is about 77 cSv, which exceeds the annual allowable of 50 cSv. For a 560-day stay on Mars, the cumulative 95% CI dose is about 120 cSv. This would exceed the career allowable dose for most females and younger males. The 95% CI dose from a major SPE would exceed the 30-day allowable dose. The probabilities of encountering a large SPE are ~3.4% for a 4X 1972 SPE and ~28% for a 1X 1972 SPE for 560 days on the surface during Solar Maximum.