The obvious answer is a compass. I have an old (1952) transit that has a compass which seems to be accurate to 15'. However, that doesn't seem too incredibly accurate at all. Furthermore, most modern digital theodolites I see don't appear to have a compass (correct me if I'm wrong). I'm curious how a surveyor actually finds north with high accuracy. Are there digital theodolites or other instruments with compasses? Are north-south lines set out with GPS? I'd appreciate any insight.
Gyrotheodolites are an older, but accurate way to find true north. They can be used on the Earth's surface, but these days they tend to be used for surveying underground, in tunneling and mining applications. GPS surveying methods are replacing gyrotheodolite methods for this application on the Earth's surface.
Gyrotheodolites are "surveying instruments composed of a gyroscope attached to a theodolite". There is more information about gyrotheodolites here.
When used underground, they are the best way to establish the initial surveying reference points underground. Depending of the extend of the underground infrastructure, occasionally, gyrotheodolites are used for check surveys of the underground reference points. After the initial reference points have been established conventional theodolites or total stations are used underground.
For underground infrastructure that can only be accessed by vertical shafts they are the best way to establish surveying reference points underground. The other way is to drop two plumb bob lines down the shaft and get the lateral co-ordinates of the plumb bob lines on the surface from known surface survey stations. Leaving the plumb bob lines untouched surveyors go underground by another shaft and survey the two plumb bob lines from underground and then survey what will be the reference point for that level underground. After making corrections for the sway of the plumb bob lines the lateral co-ordinates for the underground reference point can be calculated with good accuracy. Once that is done knowing the direction of north, underground, becomes possible.
Tunnels that connect with the Earth's surface via an adit, or similar portal, and are horizontal or sub-horizontal can be surveyed using conventional theodolites or total stations, with the first reference points taken directly from the surface via line of sight.
One limitation of gyrotheodolites is they can't be used at either the North or South Pole "where the Earth's axis is precisely perpendicular to the horizontal axis of the spinner and the meridian is undefined". Because of this gyrotheodolites tend not to be used within 15 degrees of either pole.
For surveying on the Earth's surface GPS surveying equipment can be used such the GNSS system provided by Trimble (not advertising, just using it as an example).
Depends on the accuracy you are looking for:
Solar observation. Traditional astronomical way... just place or use something very high (tower, column...) where you can know, exactly, which one is the base. F.e., on a cathedral tower, you can consider the top of it as you can know where you have the projection on the ground. Then follow the shade during the day... when you get the shortest length, it is pointing you North or South. Depends on the Hemisphere.
Star observation. Just follow one star with a theodolite or astrolab in different moments, until you get which ones are the maximum readings of it (as it is showing the limits of the star rotation against the Earth). Traditionally, the pole star was used as the declination (+89°19′8″) was very high, so the movement is very short. With a few readings, you will get the limits of the movement (by angles) and the center of it is the North. On the South Hemisphere, Crux is used.
You can use software like Stellarium to know at what exact time the sun will cross the meridian. At that time, you should cast a straight shadow on the terrain using a vertical pole. This line (the shadow) will go true North-South. In fact, you can know the angle with respect to true North of any straight pole shadow at any time of day. With that information you can figure out where is true north to decent accuracy, if you don't trust your compass.
Other way of determining true North would be at night, using what's called a polar scope. These have a reticle that shows the aparent orbit of Polaris (Northern hemisphere) and 4 stars of Octans (Southern hemisphere). You just need to point and align the reticle with the stars.
If that's not enough, you can use Plate Solving. I'm not too familiar but I remember it involves taking long-exposure photographies of the stars usign a star-tracker mout and adjusting your azimut and altitude until no star trails are present. The longer the exposure, the greater the accuracy of true north. You will need a software to analize the photographies and solve the star positions using a catalogue.
I just purchased a Bosch 32 expecting that it would be able to find north but to no avail. It's just a really fancy paper-weight. The US Army Field Artillery uses either the M2A2 Aiming Circle or the GLPS (GUN LAYING AND POSITIONING SYSTEM) to find direction in order to lay the howitzers on an azimuth of fire. The M2A2 is not GPS capable but has the ability to find north based on its declination constant. The GLIPs is GPS Capable. Both systems use a Position Azimuth Determining System in order to receive its information. It's unfortunate that you cannot purchase these items else I would.