Many ICs have input protection diodes that will become conductive when the input falls outside the bounds of the power supply potentials (one or the other diode becomes forward biased), having a voltage "clamping" effect (left):

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R1 is usually inserted to prevent a lot of current flowing during such clamping. This also prevents diode forward voltage from excceeding 0.3V or so, keeping the actual potential applied at the IC input to between \$-0.3V\$ and \$V_{SUPPLY}+0.3V\$. When input potential is within bounds, the diodes are reverse biased, passing negligible current as if they weren't even there. The IC's high input impedance means that resistor current is negligible, with no appreciable potential difference across it, so the IC "sees" the potential at IN.
If your device doesn't have those diodes, or the datasheet doesn't mention them, nothing's stopping you from doing this yourself, with your own diodes (shown above right).
Sometimes you wish to avoid dumping current into the positive supply rail, preferring that it go to ground instead. For that you can use a zener diode to clamp to ground:

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Above left, a single zener diode prevents the voltage at its cathode from rising above 3V, by conducting strongly in that condition. It is otherwise a regular diode, so it also prevents its cathode from becoming more negative than -0.7V. On the right we add a schottkey diode to clamp closer to zero, at -0.3V. In both cases, clamp current is directed to ground, and won't interfere with the positive supply in any way.
If for some reason the IC's input must not go negative at all, you can use the above zener diode trick, but against the positive rail, or you can clamp against something other than the supplies. Both of the following examples will allow OUT to approach, but not reach, 0V:

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The diode clamping solutions above all require a resistor to limit clamp current, but that might be a problem. For instance, reverse-biased diodes leak some current, which must traverse the resistor, causing a small voltage drop across it. That's a big problem if, say, you're trying to measure a potential with millivolt precision, using a 16 bit ADC.
It may be necessary to avoid clamping altogether. You can use resistor potential dividers to scale and offset the source, producing a potential compatible with the ADC:

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Sometimes you want as little impedance as possible between the source of signal you are measuring, and the input to the ADC. Some ADCs, particularly high speed or high resolution ones, can have appreciably low and variable input impedance. This will draw current via any impedance in the source, resulting in a voltage drop across that impedance, messing up your measurement.
A simple potential divider on its own may not be an appropriate source for direct connection to an ADC, because it becomes an additional impedance between the original signal source and the ADC itself. However, using an op-amp, with negative feedback, you can produce a near-zero impedance signal, suitable for direct connection to even the most troublesome ADC input, while simultaneously scaling and offsetting the signal:

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Alternatively, you can buffer a signal with high source impedance (such as the output of a potential divider) using a voltage follower:

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OA2 is a FET input type, with extremely high input impedance (1012Ω) and insignificant input bias currents (a few picoamps), which won't load the potential divider output at all. Crucially, it has near-zero output impedance (due to negative feedback), and will guarantee the ADC sees the signal you want it to see (op-amp input offset voltage notwithstanding).
As a bonus, by powering the follower with supplies of 0V and +3.3V, since the op-amp is physically incapable of outputting anything outside that range, you have an additional layer of security for your expensive ADC.
Update
I neglected to show you one of the most useful clamps I've come across, for signals under 400mV. It uses a diode-connected bipolar transistor, instead of a regular diode, to clamp signals to under 700mV. Where a diode will begin to conduct significant current at 200mV or so, the transistor wont begin to conduct until the signal reaches 400mV or so:

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A diode-connected transistor is just like a diode, but with much more "ideal" behaviour. As such, you can connect them in series to produce maximum forward voltages of any multiple of 0.7V. You could also use a PNP transistor, in the same configuration, to clamp negative-going signals to ground:

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Update 2
I recently answered another question, in which I used yet another technique which is worth mentioning here. It allows you to specify quite precisely an arbitrary upper limit, using a relatively high impedance reference voltage, that you could derive using a potential divider, for instance:

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Beware of reverse-biasing Q1's base-emitter junction though! This solution really only works for clamping to 5V or less, since a 0V input would reverse-bias that junction enough for it to begin conducting.