Notepadqq is similar to Notepad++ and runs natively on Ubuntu.
I recommend Notepadqq for Ubuntu users who like Notepad++.
The stable version is provided by this official PPA and you can install it by running:
sudo add-apt-repository ppa:notepadqq-team/notepadqq
sudo apt update
sudo apt install notepadqq
Then you can run it from Programming → Notepadqq in your menu, or use the notepadqq
command. (It should also appear in your Open with menu when you right-click a file in your file browser, though this may depend on your setup.)
Notepadqq is a Qt-based GPLv3-licensed FOSS text editor. Aside from running Notepad++ with Wine (which is pretty easy) or undertaking the work of porting it to GNU/Linux with Winelib (which is hard), Notepadqq is probably the most similar editor to Notepad++ that can be run on Ubuntu. It feels much like a port of Notepad++, even though it does not actually contain any Notepad++ code. As the FAQ explains:
Is this project affiliated with Notepad++?
No. This project is independent from Notepad++ and doesn’t use any of
its code. Notepadqq is inspired by Notepad++: this means that we try
to reproduce its best traits, while not being afraid to make different
choices on what we think can be improved.
On my Lubuntu 16.04 LTS system, I have both Notepadqq and Notepad++ installed. (Notepad++ runs with Wine.) I had no problems installing either and they both work reasonably well, though Notepad++ is sometimes slow when I use it to edit very large source code files. I haven't had this problem with Notepadqq, nor with Notepad++ in Windows. Note that I haven't benchmarked them, nor have I extensively used both on the same system, so please take my remarks about performance with skepticism.
The two editors do look somewhat different, but you can see that they are largely similar. I find the actual experience of interacting with them to be almost the same.

Notepadqq (left) and Notepad++ (right) on Lubuntu 16.04 LTS, viewing sqlite3.c
.
You can further adjust their appearance with themes (they are both theme-able) and by adjusting your desktop environment's settings. They may not look exactly like this on your Ubuntu system. I use the LXDE desktop environment. If you use another desktop environment, such as GNOME, the window borders and title bars will be styled differently.
As an example of how Notepadqq differs from Notepad++ but not by all that much, and of how it is easy to use if you already know your away around Notepad++, here's how they both look when you search an open file using a regular expression:

Regex search in Notepadqq (left) and in Notepad++ (right).
Notice that they support the same three kinds of search patterns (named differently).
If you want to look at Notepadqq's source code, or compile the latest development version, see Notepadqq on Github. For instructions on how to build it from source or install official pre-built binaries on other OSes, see the README file. You may also be interested in the guidelines for contributing.