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Things that you should NOT delegate are those with direct negative (or even disasterous) impact on your responsibiity/accountabbility in front of your superiors. Let me give you an example ... how do you feel towards giving your personal/professional e-mail id and password to anyone ??? even the ones who you delegate other things to ?
How would explain/justify abuse of this e-mail id/password in case your delegate decides (for whatever reason) to send from your e-mail something which is totally unacceptable/untolerable ??
How would you feel about giving a signed blank checque to anyone ?? Would you consider such a thing for your delegate ??
To draw some conclusions here , you may delegate work items, actions, tasks that usually fit within the day-to-day operations, not the ad-hoc situations , not things related to financial or personal liability on you or on the company.
It is also not recommended to delegate things of signatory authority for major impact / critical decisions, hiring /firing people, promotions and career advancement .