Under UK law, if an employee makes an invention as part of his standard employment, then unless his contract of employment says anything to the contrary, the invention belongs to the company. However, the inventor must be identified as such in the patent and confirm how the company acquired the rights to the invention.
If an individual inventor or group of inventors created an invention outside employment, then the inventor belongs to the inventor or inventors.
Similarly with a design, if the design was created in the course of normal work as an employee, the design belongs to the company. If the design if created outside employment, the design belongs to the designer.
Design right belongs to a commissioner, unless otherwise stated in the contract of commission, whereas copyright belongs to the creator and not the commissioner unless otherwise stated in the contract of commission.
Ownership of any form of intellectual property can be assigned from one entity to another. Where the intellectual property right is a registered right, then in order for the assignee to use his rights, the assignment must be registered at the body granting the right.
Under US law, an invention is the property of the inventor. A contract of employment can stipulate that the employee must assign an invention to the company, but the application must always be initially filed in the name of the inventor(s).