First thing you need is something to convert the current to a voltage. That is... a resistor (V = I*R). If the current sense resistor has a superimposed voltage (a "high side sense"), you will need a differential amplifier as well, such as a current sense amp (there are purpose-made ICs for this), or an instrumentation amplifier.
Some sort of EMC protection and filtering, and acquisition bandlimiting, is almost certainly required. Normally, you'd have a series resistor between the current-sensing shunt and the circuit. Then some clamp diodes (e.g. BAV99) to the supply to contain ESD and spikes. Then some capacitance (and subsequent RC/LC or active op-amp filtering) to limit bandwidth, further contain outside surges, and make things nice and smooth for the ADC readings.
The cutoff frequency of all that filtering is generally on the order of the sample rate. It doesn't have to be below half (i.e., the Nyquist frequency). There are very good reasons why you might want it above the sample rate. Control loops (minimal latency, frequency-domain aliasing not a concern) are one, equivalent-time sampling (intentionally using aliasing to reconstruct a faster signal) is another. If you do want to reconstruct the bandlimited signal (i.e., satisfying the Nyquist Sampling Theorem), you need a filter that excludes everything* above Fs/2.
*How much depends on accuracy; 40dB attenuation is 1% error (due to possible aliasing), and etc. You can use a crappy filter, pretty far from Fs/2, to achieve this; or a pretty sharp filter, relatively close to Fs/2 (usually Fs/3 to Fs/4, with a 3-7 order filter -- depending on filter type), to maximize bandwidth or minimize step response (while retaining "real time" (i.e. no aliased or equivalent-time-sampling phenomena) response).
Tim