The definition of a carcinogen is simply 'A substance that causes cancer or is believed to cause cancer'.
This definition is somewhat ambiguous when it comes to assesing the risk of working with substances, as the 'belief' clause means full safety measures should be taken even if the risk is unproven.
There are also different means of categorising carcinogens according to the risk they present. For instance the current American Conference of Governmental Industrial Hygienists (ACGIH) uses the following :-

A1 - Confirmed Human Carcinogen: The agent is carcinogenic to humans based on the weight of evidence from epidemiological studies of, or convincing clinical evidence in, exposed humans.

A2 - Suspected Human Carcinogen: The agent is carcinogenic in experimental animals at dose levels, by route(s) of administration, at site(s), of histologic type(s), or by mechanism(s) that are considered relevant to worker exposure. Available epidemiological studies are conflicting or insufficient to confirm an increased risk of cancer in exposed humans.

A3 - Animal Carcinogen: The agent is carcinogenic in experimental animals at a relatively high dose, by route(s) of administration, at site(s) of histologic type(s), or by mechanism(s) that are not considered relevant to worker exposure. Available epidemiological studies do not confirm an increased risk of cancer in exposed humans. Available evidence suggests that the agent is not likely to cause cancer in humans except under uncommon or unlikely routes or levels of exposure.

A4 - Not Classifiable as a Human Carcinogen: There is inadequate data on which to classify the agent in terms of its carcinogenicity in humans and/or animals.

A5 - Not Suspected as a Human Carcinogen: The agent is not suspected to be a human carcinogen on the basis of properly conducted epidemiological studies in humans.

As you can see the hazard and risk of working with a substance, with regards to it's carcenogenic properties, can be ranked from 'high' to 'none'. If you are working with a material that does not come with relevant hazard data, you should ask your employer to obtain it, and use all necessary safety protocols.

In the USA the OSHA - Occupational Health and Safety Administration produces a paper the 'Hazardous Communications standard 1910.1200' to define the action employers should take to protect employees from any risk due to exposure. The most relevant sections of this document, with respect to carcinogens are as follows :-

(d)(4)
Chemical manufacturers, importers and employers evaluating chemicals shall treat the following sources as establishing that a chemical is a carcinogen or potential carcinogen for hazard communication purposes:

(d)(4)(i)
National Toxicology Program (NTP), "Annual Report on Carcinogens" (latest edition);

(d)(4)(ii)
International Agency for Research on Cancer (IARC) "Monographs" (latest editions); or

(d)(4)(iii)
29 CFR part 1910, subpart Z, Toxic and Hazardous Substances, Occupational Safety and Health Administration.

Note: The "Registry of Toxic Effects of Chemical Substances" published by the National Institute for Occupational Safety and Health indicates whether a chemical has been found by NTP or IARC to be a potential carcinogen.

(d)(5)
The chemical manufacturer, importer or employer shall determine the hazards of mixtures of chemicals as follows:

(d)(5)(i)
If a mixture has been tested as a whole to determine its hazards, the results of such testing shall be used to determine whether the mixture is hazardous;

..1910.1200(d)(5)(ii)

(d)(5)(ii)
If a mixture has not been tested as a whole to determine whether the mixture is a health hazard, the mixture shall be assumed to present the same health hazards as do the components which comprise one percent (by weight or volume) or greater of the mixture, except that the mixture shall be assumed to present a carcinogenic hazard if it contains a component in concentrations of 0.1 percent or greater which is considered to be a carcinogen under paragraph (d)(4) of this section
A carcinogen is anything that induces cancer. Carcinogens are usually thought of as chemical in nature, like benzene or tobacco smoke, but may also cause cancer through physical action, as with ultraviolet light or asbestos. Carcinogens do not tend to fall into deeper categories -- just as there are hundreds of kinds of cancer, there are hundreds (or thousands, or more) of vectors that may cause it to start.

Determining what agents are and are not carcinogenic is done by committee, through sort of an ad-hoc voting system. Two committees are responsible for this determination: internationally, the International Agency for Research on Cancer (IARC), part of the World Health Organisation; and in the US, the Annual Report on Carcinogens done by the National Toxicology Program (NTP).

The IARC format is more widely used, and divides carcinogens into three categories. Category 1 carcinogens are those which have a determined relationship with cancer in humans; confirmed human carcinogens. Category 2A includes agents which have less evidence in humans, but are unequivocally confirmed carcinogens in animal studies; probable human carcinogens. Category 2B is for substances which have some evidence from animal experimentation, and are considered good candidates for further study; possible human carcinogens. Finally, there is Category 3, which I don't know much about except that their evidence for acceptance is considered "limited."

The NTP format is easier, with only two categories. These definitions are both copied directly from the NTP report. A, profiles for agents, substances, mixtures or exposure circumstances known to be human carcinogens. And B, profiles for agents, substances, mixtures or exposure circumstances reasonably anticipated to be human carcinogens. While the NTP report and format are less accepted by the world, it should be noted that the NTP leads the world in experimentation to determine toxicity and carcinogenicity.

Here are some of the more interesting and unexpected carcinogens from the lists:

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