Basic Task Resolution; aka, Skills.

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Choose the skill you wish to use, add it's level to the governing attribute, and roll 1d10.
If you roll a 10, you may roll 1 more time and add it to your total.
If you roll a 1, then you have Fumbled.  Bad things may happen as a result, depending on the task

Some potential modifiers to the target number include

Complex Repair +2
Don't have the right parts +2
Don't have the right tools +3
Drunk, drugged, or tired +4
Hostile environment +4
Very hostile environment +6
Area is lit too brightly +3
Insufficient Light +3
Pitch Blackness +5

Advancement.  Cyberpunk doesn't have levels, and doesn't have Experience Points.  Instead, it is skill based.  As you perform tasks, your skills will slowly improve, which marks your character getting better

This is measured by IP; Improvement Points. 

The first level of a new skill costs 10 IP.  From then on, in order to improve your skill, take your current level and multiply it by 10, and this is the IP cost to go up by +1 (a skill of +2 costs 20 IP to advance to +3).

Some skills are more difficult than others, and have another multiplier.  The skill lists sometimes lists numbers in ( )'s, this is the IP multiplier.  Basic Tech for example is (2), so to improve from +2 to +3 would cost 40IP (current level 2 x 10 x 2 IP multiplier)

There are 3 main ways to raise a skill

Study/Practice.  You get a book, watch a video, try to train yourself.  In general, 1 day of Bookleaning is worth about 1 IP, and you can only improve your skill up to a +2 in this fasion

Being Taught.  You find yourself a teacher.  In general, this is better than self teaching.  However, the teacher must have the skill higher than your current skill (They can't teach you if you know more than them), and they must also have the Teaching skill.  The teacher then averages his Teaching skill, and the skill you wish to learn, and this is as high as he can teach you.  Example, Artemis is teaching you how to shoot a handgun.  He is very good, with a Handgun skill of +8, but he is not the best teacher, Teaching +2..  the highest he can teach you is (8+2 / 2) +5.  The amount of IP granted varies based on the length of the lesson, and perhaps other extraneous situations, but generally ranges from 1-5 IP per lesson.

Experience.  The best teacher.  When you do something very well, complete a difficult task under pressure, use a skill at a dramatically appropriate moment, etc, you may be awarded with IP on the spot for that skill.

There is no specific place on the Sheet for skill IP, so I recommend using the blank spot next to the skill level box.

There is, however, a space on the character sheet for General IP.  This is a general pool that can be used for any of your skills.  Maybe you are getting close to advancing your Melee skill, but never had a chance to use it this session for some specific IP.. go ahead and finish it off with some IP from your general pool.  It is recommended to only spend it to advance skills that were used during the session, and also not raising a skill by more than +1 per game session.

"Five million Cybermen, easy. One Doctor? NOW you're scared!" - Rose Tyler