1NT (15-17)

The modern, strong 1NT opening shows a balanced hand with 15-17 HCP. In the old days, 1NT showed 16-18 HCP, which has fallen out of favor due to its lower frequency.

Examples of hands suitable for a 1NT opening:

A873
AJ42
AJ2
KT
     K97
T2
K87
AKQ43
     AJ
876
AK98
KT32
     AKQ
KQJ2
J65
973

Shape

1NT promises balanced suit distribution without any singletons or voids, namely:

4-4-3-2
4-3-3-3
5-3-3-2

However, 1NT may also be opened on semi-balanced distribution:

2-3-3-6 (a long minor)
2-2-4-5 (excluding length in both majors)

In "Commonsense Bidding", Bill Root cites the following semi-balanced hand as an example of distributed strength suitable for 1NT:

With a 5-Card Major

The SAYC bidding system states that 1NT "can be made with a five-card major". This is up to partnership agreement.

The advantages of opening 1NT with a 5-card major:

It's slightly more preemptive. It's riskier for the opponents to overcall 2 over 1NT than to overcall 1 over 1.
It's more descriptive. Your point range and balanced shape are immediately shown, whereas a 1/ opening bid may be made on all kinds of hand patterns and strengths.

The disadvantage of opening 1NT is the danger of missing a major-suit fit. This can happen whenever partner holds 3- or 4-card support:

She passes 1NT with a weak hand.
She passes a 2-level overcall. Now you must decide whether to introduce your major at the 2-level.
She makes a Jacoby Transfer when you only have doubleton support.

Consequently, partnerships may have special agreements with a 5-card major (up for discussion):

Opening 1NT with 3 cards in the other major, in case partner makes a transfer bid.
Opening 1NT with evenly distributed high-card strength.
Opening 1NT with 5 hearts. With 5 spades, consider opening 1.

Responses

The following responses are consistent with the SAYC bidding system, which is comparable to any generic 5-card major system. However, variations always apply and these responses should not be considered universal for all bidding systems.

Pass
0-7 points, no 5-card major (use a Jacoby Transfer instead) and no 6-card minor (bid 2 to relay to your minor suit).
The 2 Response
Stayman, showing 8+ points and at least one 4-card major. However, it typically excludes hands with 4-3-3-3 shape.
The 2 Response
A Jacoby Transfer, showing 5+ hearts.
The 2 Response
A Jacoby Transfer, showing 5+ spades.
The 2 Response
Artificial, showing a weak 6+ card minor suit. Opener is required to bid 3. Responder will pass or correct to 3.
Minor-Suit Stayman is a popular non-SAYC alternative to this approach.
The 2NT Response
Invitational to 3NT. Shows 8-9 HCP and balanced shape. Denies a 4-card major unless holding 4-3-3-3 shape.
The 3 Response
6+ clubs, 7-8 HCP. Invitational to 3NT.
As an alternative approach, it may simply be a weak sign-off bid. This method is espoused by expert Richard Pavlicek, among others.
The 3 Response
6+ diamonds, 7-8 HCP. Same comments as 3, above.
The 3 Response
6+ hearts with slam interest. However, this agreement rarely comes up. Many players would simply start with a Jacoby Transfer.
The 3 Response
6+ spades with slam interest. Same comments as 3, above.
The 3NT Response
Natural, 10-13 HCP.
The 4 Response
Gerber, an artificial ace-asking bid.
The 4NT Response
Quantitative, inviting 6NT.

In Competition

If the opponents double 1NT, then Stayman and Jacoby Transfers are still "on".
If the opponents overcall 1NT, then Jacoby Transfers are "off". However, Stayman may still be possible via a cuebid.
If the opponents double a 2/ Jacoby Transfer, then opener is allowed to pass with only 2-card support.

As a non-SAYC alternative, the Lebensohl convention can be used to show a variety of hand types if the opponents double or overcall through 2.