Is my landlord allowed to enter my apartment without notification?

This largely depends on the laws where you live. I can pretty well promise you that "A police matter" is NOT the sole reason a landlord can enter your apartment.

Here in Ontario, a Landlord IS permitted to enter your apartment WITHOUT notice if he has obtained a Bench Warrant to do so and if there is an Emergency situation; IE: Your on Vacation (Or even just at work) and a pipe bursts under your sink, the Landlord is WELL within his rights to enter your apartment with no notice to you what so ever. As well, if you have given notice to a Landlord of your intention to move out, the Landlord is then able to enter your apartment for the purposes of evealuating it as well as to show it to future tenants, however, it is law that they may only do this between 8am and 8pm, and it is generally accepted that they will not just walk in, but knock and see if this is a good time.

Other then this, a Landlord may not enter your Apartment with out proper notice. Generally speaking, at least 24 hours notice is required as "proper notice". In fact, if you request him to do a repair to your apartment, he either has to wait until you are around, or have you sign an agreement allowing him access to your place during an agreed upon period of time.

As stated previously, this depends on your State's laws. Here in Wisconsin we must give our residents a 12 hour notice that we will be entering their apartment. This does not include times when there may be an emergency such as water leaking, fire, etc.

In Massachusetts, a landlord is allowed to enter a premises without notice for a number of reasons. These reasons are usually specified in the lease agreement. Otherwise, the landlord must give "reasonable" notice. What reasonable notice is, is open to interpretation.

Some of the reasons I can enter my tenant's apartment without notice as specified in MY lease agreement:
a) If it appears the apartment has been abandoned
b) If I feel that the life or safety of the tenant is at risk
c) If I feel that there is a risk of damage to the premises

If you are having a problem, definitely read your lease agreement. If you do not have a written lease, this would be governed by the laws of the state you live in. I'm assuming that other countries have similar tenant protection laws.

Only in a police matter.

If you have issues with such things, http://ratemyapartments.com is a good place to report them. This way others will be aware before stepping into the same mistakes.

Also i think that it must be stated in the lease/rental agreement. If it is in there and you have signed it they can go in at any reasonalble time.

The 1st. answer is correct. This is a matter of State law.

In an emergency, yes, and if s/he can show efforts were made to contact you first but to no avail.