requirements will vary depending where you or your clients are. they can be imposed through regulations but there can also be things that are not regulated but clients still request them.
in general safety standards are country specific although robotic safety standards are harmonized and quite similar.
companies in same country can have different internal standards. so there is no universal answer, specially since you did not state what country you are from.
many small companies are happy that they got new cell that does the work and do not push for more (documentation etc.).
automotive companies in particular are very aware of this and will usually spell everything exactly as part of requirements long before project starts. they will not be happy with cell being operational, they will insist on full documentation, source code of everything, and will also dictate what products can be used - you cannot just go and pick robot or PLC or HMI that you are familiar with or can get good deal on. They will usually dictate every little thing including what type of wiring labels to use, how code must be structured etc. Often they will have ready templates or project examples.
I am in Canada so here there are number of standards like
Z432 - Machine safety directive
Z434 - Robotic safety
Z142 - Press safety
etc.
Then there are electrical standards (panel building , wiring, power distribution,...), lockout etc,
Then there are field inspections and approvals. If product does not meet CSA or UL it cannot be used and must be replaced or modified to meet the standards. This can be expensive.
Also there is a pre-health and safety review that goes over hazards and evaluates risk and remedies including determining minimum acceptable safety category.
I am sure similar exists pretty much everywhere but you need to ask around in place where you live.
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I am from Ukraine and i work on European factory. We are adhere to Ukrainian, European and internal standards. I plan to develop an internal standard for the integration of industrial robots, to do this, I need to find all possible world standards and make one common which will take into account everything, especially the points of robot safety.
As far as I understand from what I found on the internet, the basic standards for robots in Europe are:
- EN 12100
- EN ISO 10218-2
I am interested in requirements that are not spelled out in the standards, but customers still require them, as in your example with automotive companies. (Even if these requirements seem unnecessary)
For example:
- What package of documents usually provide integrator after
installation?
- Do customers looking for certified integrator who have completed
certain courses for programming or it is usually not necessarily?
- What types of security are commonly used? Light
barrier/fence/other?
- As usually used robot independently or with PLC? (As far as I know,
robots have their inputs and outputs which can be used for simple
projects)
- Which manufacturer of external axes is usually chosen? Same as
robot manufacturer for cell or other company?
- Are there any requirements for the robot tool?
- Are there any requirements for the programmed speed of the robot?
- What data must be obtained in order to easily maintenance and
adjust the program (if we need it in future)?
- Who usually develops the concept for integration? Customer or
integrator?
- Etc.
Also what need to pay attention to after installing and configuring the robot. What need to check?
I understand that need to check some basic things such as:
- Security system operation
- Integrity of equipment
- Full movement of the work axes and stop at the limits
- Complete set of a cell
What else can be checked?
I understand that these questions may seem simple to you which do not require explanation, but remind I have just started to study this topic and all this is still new to me. 
Thank you.