How do materials pass in and out of a cell nucleus?

The nucleus of a cell is enclosed in a membrane. The membrane acts as a barrier to passage into the nucleus of the cell, but is selectively permeable. The membrane around the nucleus of a cell, otherwise known as the nuclear membrane, is impermeable to larger molecules but allows for the regular passage of smaller molecules.
The nuclear membrane is also known as the nuclear envelope and is made up of two cellular membranes arranged parallel to one another. The membrane encircles and completely contains the genetic material within the nucleus of the cell.
Access to the nucleus is achieved with the aid of nuclear pores, which are regularly arranged on the surface of the nuclear membrane. These pores allow the free passage of smaller molecules and ions while they prevent larger molecules such as nucleic acids and proteins from randomly entering the nucleus. These larger molecules must be transported into the nucleus of the cell through the nuclear pores, a process that is mediated by a number of factors.
The nuclear envelope essentially controls access to the nucleus of the cell, only allowing larger molecules like DNA and proteins into the nucleus during processes like gene expression.