Necessity of Cyclic Prefix in OFDM
In CP-OFDM, the transmitted signal is the output of the IDFT appended with a cyclic prefix. No pulse-shaping is used.

In so-called "new waveforms" proposed as candidate for 5G, pulse shaping is heavily used.

In FBMC, each subcarrier (i.e each complex exponential) is convoluted with a pulse with a certain shape (usually a bell shape with beginning and end at zero amplitude). As a consequence of convolution, the amplitude of each filtered subcarrier at the beginning and at the end of the multicarrier symbol goes to zero. 
Consequently, in case of multi-path, this zero-valued beginnings and ends avoid ISI, since the part of the previous symbol overlapping most with the current one is its end, which as we said is zero-valued (or almost) and the current symbol has little energy at its beginning (zero or almost).

Hope this helps.