How does one make a Zero Latency, Linear Phase, or Natural Phase filter?
To add to hotpaw's answer, Linear Phase filters can only be implemented with all zero structures (FIR Filters) but they can be approximated with IIR filters such as the Lerner Filter (https://ieeexplore.ieee.org/document/1444798) and other equiripple phase approaches. In order to achieve linear phase, the coefficients of the filter must be either symmetric and assymetric as given in the examples further below.

Zero Latency filters usually refer to Zero-phase digital filtering which is strictly a post processing technique by processing the data through the filter in both forward and reverse directions such that the phase in each direction is cancelled. The filter itself has a transfer function that is the square root of the desired since the signal is filtered twice.  For more information on zero-phase digital filtering, see the filtfilt command available in Matlab/Octave and Python scipy.signal. This is not actually a filter with zero processing delay, but given an input and output data set, the two will be perfectly aligned in time as if the filter had no delay if you later compared the data set once processed (hence post-processing). It will take real time to get the first sample out of the filter that can be placed on a plot with t=0 with the input data. This is very useful for analysis in applications where you would otherwise need to compensate for the filter delay, so worth mentioning here. 

With regards to Linear Phase Filters, The following are examples that demonstrate "symmetric" and "asymmetric" taps that is a requirement to implement a linear phase filter. 

Linear Phase Type 1 FIR (Symmetric, N odd):

[ 1 2 3 2 1 ] 

Linear Phase Type 2 FIR (Symmetric, N even):

[ 1 2 3 3 2 1 ] 

Linear Phase Type 3 FIR (Asymmetric, N odd):

[ 1 2 0 -2 -1 ]     

Linear Phase Type 4 FIR (asymmetric, N even):

[ 1 2 3 -3 -2 -1 ]

Note importantly how the center bin MUST be zero in order to achieve the required asymmetry for the Type 3. The structures above for each type restrict the zeros of the filter to always be on $z = -1$ for Type 2, and $z = \pm 1$ for Type 3 and $z=0$ for Type 4 restricting which applications can be used for each type. 

The following plot further demonstrates this and shows the purpose of having the FIR linear phase "Types":