I wanted to update curl, but the version available on 16.04 of Ubuntu was 7.47.0, (not the latest 7.57.0) so the nice apt-get method wasn’t an option.
Download the latest release from http://curl.haxx.se/download.html to your home directory:
wget http://curl.haxx.se/download/curl-7.57.0.tar.gz
Install the tools to compile this release:
apt-get install libtool
apt-get install make
Unpack the source code:
tar -xvf curl-7.57.0.tar.gz
Compile:
cd curl-7.57.0 # enter the directory where curl was unpacked
./buildconf
./configure --with-ssl
make
make install
Update the system’s binaries:
mv /usr/bin/curl /usr/bin/curl.bak
cp /usr/local/bin/curl /usr/bin/curl
You are done 
! The following will display your version of curl.
curl -Vv
If you are using this in PHP, you should restart:
service apache2 restart # if using apache
service php7.1-fpm restart # depending on your version
Issues
I should add my initial run of this did not produce a curl that could handle https (!) or handle some common encodings. The first issue was solved with:
sudo apt install libssl-dev
and appending –with-ssl to the ./configure command used in compiling above.
./configure --with-ssl
The second through downloading zlib and compiling:
wget http://www.zlib.net/zlib-1.2.11.tar.gz
tar -xvf zlib-1.2.11.tar.gz
cd zlib-1.2.11/
./configure
make
make install
I then continued with the initial install from the compile section, using:
./configure --with-ssl --with-zlib
This time I could confirm they were both installed by having the https protocol present, and a zlib version displaying on the first line of:
curl -Vv